Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
For $2x+2y-3z\geq0$ and similar prove that $\sum\limits_{cyc}(7z-3x-3y)(x-y)^2\geq0$
Let $x$, $y$ and $z$ be non-negative numbers such that
$2x+2y-3z\geq0$, $2x+2z-3y\geq0$ and $2y+2z-3x\geq0$. Prove that:
$$(7z-3x-3y)(x-y)^2+(7y-3x-3z)(x-z)^2+(7x-3y-3z)(y-z)^2\geq0$$
I have a proof for the following weaker ineq... | We can use the substitutions $2x+2y - 3z = u, \ 2y+2z - 3x = v$ and $2z+2x - 3y = w$ for $u, v, w \ge 0$.
By solving the system of equations above, we get
$x = \frac{2u+v+2w}{5}, \ y = \frac{2u+2v+w}{5}$ and $z = \frac{u+2v+2w}{5}$.
The rest is the same as @yao4015's solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2181568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integration of $\int_0^a \frac{\log\left|x^2 - 1\right|}{1 - x^2} dx$ I am trying to integrate something and after some changes of variable and an integration by parts I am stuck with this integral:
$$\int_0^a \frac{\log\left|x^2 - 1\right|}{1 - x^2} dx$$
I know that $a < 1$.
My question is twofold:
*
*Is it legal, ... | After some hours and a change of variable more I have been able to
'solve' the integral, although its result is in terms of the Polylogarithm,
as some of you have pointed out.
I was suggested in Twitter to try the expansion $log(1-x^2) = log(1-x) + log(1+x)$
and then the change $1-x = \exp(t)$ in the first case and $1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2183734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Incorrectly solving the determinant of a matrix Compute $det(B^4)$, where $B =
\begin{bmatrix}
1 & 0 & 1 \\
1 & 1 & 2 \\
1 & 2 & 1
\end{bmatrix}
$
I created
$C=\begin{bmatrix}
1 & 2 \\
1 & 1 \\
\end{bmatrix}
$ and
$
D= \begin{bmatrix}
1 ... | Why did you do that? Just use determinant along the first row. This way is easier because there is a 0 in that row.
$$
\begin{vmatrix}
1 & 0 & 1 \\
1 & 1 & 2 \\
1 & 2 & 1 \\
\end{vmatrix}=1\begin{vmatrix}
1 & 2 \\
2 & 1 \\
\end{vmatrix}-0\begin{vmatrix}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Evaluating limits using taylor expansion $$\lim_{x\to 0^{+}} (\ln x)^3\left(\arctan\left(\ln\left(x+x^2\right)\right) + \frac{\pi}{2}\right) + (\ln x)^2$$
I have this limit in my sheet and the answer is $\frac 13$.
But I don't know how to approach any step using taylor expansion near zero (this is our lesson by the wa... | When $x\to 0^+$,
$$
\ln(x+x^2) = \ln x + \ln(1+x^2) = \ln x + o(x) \tag{1}
$$
and this goes to $-\infty$ as $x\to 0^+$. To continue, we would like to have $\arctan u$ with $u\to 0$, though. When $x<0$, we can use the fact that
$$
\arctan x + \arctan \frac{1}{x} = -\frac{\pi}{2} \tag{2}
$$
to rewrite (for $x$ small en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Computing $\int \frac{7x}{(2x+1)} dx$ I'm trying to compute the following integral $$\int \frac{7x}{(2x+1)} dx$$
Unfortunately Wolfram Alpha gives me a different result, but other integral calculators say that my result is correct. So where is my error:
$$\int \frac{7x}{(2x+1)} dx = \frac{7}{2}\int \frac{2x}{(2x+1)} dx... |
If $y=F(x)$ be a primitive of the function $y=f(x)$ then for every real $k$ the function $y=F(x)+k$ is a primitive also.
The primitive of $\dfrac{7x}{2x+1}$ is $\dfrac{7}{4} (2x -\ln(2 x + 1))$, then for every real $k$ the function $y=\dfrac{7}{4} (2x -\ln(2 x + 1))+k$ is a primitive also.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2185374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Distance between ellipse and line What is the distance between the ellipse $$\frac{x^2}{9}+\frac{y^2}{16}=1$$ and the line $y=6-x$.
I think I need to be using Lagrange Multipliers but don't know how.
| The minimum value it's the minimal distance between $x+y=6$ and $x+y=c$,
where $x+y=c$ is a tangent line to our ellipse
and since $\frac{xx_1}{9}+\frac{yy_1}{16}=1$ is a tangent line to the ellipse in $(x_1,y_1)$,
we obtain $\frac{x_1}{9}=\frac{1}{c}$ and $\frac{y_1}{16}=\frac{1}{c}$.
We can assume that $c\neq0$ bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2186411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
Relations similar to $\sin(\pi/18) + \sin(5\pi/18) = \sin(7\pi/18)$ This I checked numerically (and can be proven analytically easily). I guess there are many similar relations between the numbers
$$\{ \sin\frac{m\pi}{q}, 1\leq m \leq q \} , $$
where $q$ is some integer.
Can anyone give some reference?
| More generally, let $T_n(x) + 1$ have a factor $Q(x)$ (over the rationals) where $T_n$ is the $n$'th Chebyshev polynomial of the first kind. A root of $Q(x)$ will be $\cos(j \pi/n)$ for some integer $j$. Using trigonometric identities, we can then express $Q(\cos(\pi/n)) \sin(\pi/n)$ as a rational linear combination ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Prove that the spectrum of $K_n$ is $((n-1)^1, (-1)^{(n-1)})$ I am trying to prove that the spectrum of the complete graph $K_n$ is $((n-1)^1, (-1)^{(n-1)})$ (where superscripts denote multiplicities of eigenvalues, not exponents). I have part of the proof but having trouble completing it.
The adjacency matrix $A(K_n)$... | Hint: The $n - 1$ vectors
$$
\left\{
\begin{bmatrix} 1 \\ -1 \\ 0 \\ \vdots \\ 0 \end{bmatrix},
\begin{bmatrix} 1 \\ 0 \\ -1 \\ \vdots \\ 0 \end{bmatrix},
\ldots,
\begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ -1 \end{bmatrix}
\right\}
$$
are all in the kernel of $\lambda I - A$ for $\lambda = -1$. Why are these vectors line... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2188917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Limit without l'Hopital or Taylor series: $\lim\limits_{x \to 0} \frac{x\cos x- \sin x}{x^3}$ find the limit without l'Hôpital and Taylor rule :
$$\lim\limits_{x \to 0} \frac{x\cos x- \sin x}{x^3}=?$$
My Try :
$$\lim\limits_{x \to 0} \frac{x\cos x- \sin x}{x^3}\\=\lim\limits_{x \to 0}\frac{x\cos x \sin x- \sin x\sin x}... | If you are allowed to use the well-known limit
$$\lim_{x \to 0}\frac{\sin x}{x}=1$$
then
$$\lim_{x \to 0}\frac{\tan x}{x}=1$$
follows easily and with a bit more effort (see here), you have:
$$\color{blue}{\lim_{x \to 0}\frac{\tan x-x}{x^3}=\frac{1}{3}\tag{1}}$$
Now for your limit and using $\color{blue}{(1)}$:
$$\lim_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Find $k$ such that $\frac{1}{(1+x)^k}+\frac{1}{(1+y)^k}+\frac{1}{(1+z)^k}\geq\frac{3}{2^k}$ for $xyz=1$.
Find all $k\in\mathbb{R}^+$ such that for all $xyz=1$, $x,y,z\in\mathbb{R}^+$, we
have
$$\frac{1}{(1+x)^k}+\frac{1}{(1+y)^k}+\frac{1}{(1+z)^k}\geq\frac{3}{2^k}.$$
If we set $$x=\frac{a}{b},\,y=\frac{b}{c},\,z=... | As shown below, an easy partial result is that $k_{\text{min}} \ge 1$.
Let $k$ be fixed with $0 < k < 1$.
WIth the substitutions
$$x=\frac{a}{b},\,y=\frac{b}{c},\,z=\frac{c}{a}$$
let $a = b = 1$, and let $c \to 0^{+}$. Then
\begin{align*}
\;&\frac{a^k}{(a+b)^k}+\frac{b^k}{(b+c)^k}+\frac{c^k}{(c+a)^k}\\[6pt]
&\to \;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Does the sequence $x_{n+1}=x_n+\frac{1}{x_n}$ converge or diverge? Set $x_1=a$, where $a > 0$ and let $x_{n+1}= x_n + \frac{1}{x_n}$. Determine if the sequence $ \lbrace X_n \rbrace $ converges or diverges.
I think this sequence diverges since Let $x_1= a > 0$ and $x_{n+1}= x_n + (1/x_n)$ be given. Let $x_1=2$ since $a... | HINT: Show that $x_{n + \lceil 2 x_n \rceil} > 2 x_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Question about specific volume integral. So I need to evaluate the following volume integral, and I'm not sure which bounds to use and that makes the question kind of vague for me at least.
I've tried to solve it myself, but each try leaves my with a different solution, so I was wondering if someone could show me how ... | How do you find limits of integration?
Inside the sphere... this is asking for spherical coordinates.
$x = \rho \cos\theta\sin\phi\\
y = \rho \sin\theta\sin\phi\\
z = \rho\cos\phi$
now plug these into the equations given.
$z^2 = 3x^2 + 3y^2\\
\rho^2 \cos^2\phi = 3\rho^2 \cos^2\theta\sin^2\phi + 3\rho^2 \sin^2\theta\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove this inequality $\sqrt[3]{(a+b+c)(b+c+d)}\ge\sqrt[3]{ab}+\sqrt[3]{cd}$
Let $a,b,c,d>0$ show that
$$\sqrt[3]{(a+b+c)(b+c+d)}\ge\sqrt[3]{ab}+\sqrt[3]{cd}$$
Idea: Hence, we need to prove that
$$(a+b+c)(b+c+d)\ge ab+cd+3\sqrt[3]{(ab)^2cd}+3\sqrt[3]{(cd)^2(ab)}$$
$$\Longleftrightarrow ac+bc+bd+ad+b^2+c^2\ge 3\sqr... | It's just Holder:
$$(a+b+c)(b+c+d)\left(\frac{b}{b+c}+\frac{c}{b+c}\right)\geq$$
$$\geq\left(\sqrt[3]{a\cdot(b+c)\cdot\frac{b}{b+c}}+\sqrt[3]{(b+c)\cdot d\cdot\frac{c}{b+c}}\right)^3=\left(\sqrt[3]{ab}+\sqrt[3]{cd}\right)^3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find if $\sum\limits^{\infty}_{n=2} \frac{3}{n^2+3n}$ converges or not. If converges, find its sum. Find if $$\sum^{\infty}_{n=2} \dfrac{3}{n^2+3n}$$
converges or not. If converges, find the sum.
I used the integral test to determine whether the series is convergent or not.
I found $$3\int^{\infty}_{2} \dfrac{1}{x^2+3x... | Observe
\begin{align}
\frac{3}{n(n+3)} = \frac{1}{n}-\frac{1}{n+3}
\end{align}
which means
\begin{align}
\sum^\infty_{n=2}\left(\frac{1}{n}-\frac{1}{n+3}\right) =&\ \left(\frac{1}{2}-\frac{1}{5}\right)+\left(\frac{1}{3}-\frac{1}{6}\right)+\left(\frac{1}{4}-\frac{1}{7}\right)+\left(\frac{1}{5}-\frac{1}{8}\right)+\ldots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Using the substitution method for a simple integral I have been playing with the substitution rule in order to test some ideas with computational graphs. One of the things I'm doing is applying the substitution to well known, and easy, integrals. For example, let's use that method to find the indefinite integral for
$... | You have one mistake. If you are changing $dx$ into $du$, then you need to convert all terms containing $x$ into $u$.
So we have,
$$\int x^2 dx = \int u \frac{1}{2\sqrt u} du$$
$$= \frac 12 \int \frac{1}{\sqrt{u}}\cdot u du = \frac 12 \int \sqrt u du$$
$$= \frac{1}{2} \cdot u^{\frac{3}{2}}\cdot\frac{2}{3}+ C$$
$$= \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
proof by induction that $3^{2n-1} + 2^{2n-1}$ is divisible by $5$ I am supposed to proof by induction that $3^{2n-1} + 2^{2n-1}$ is divisible by $5$ for $n$ being a natural number (integer, $n > 0$).
What I have so far:
Basis: $n = 1$
\begin{align}
3^{2 \cdot 1-1} + 2^{2 \cdot 1-1} & = 3^1 + 2^1\\
... | let $$T_n=3^{2n-1}+2^{2n-1}$$ and $$5|T_n$$ we have to Show that $$5|T_{n+1}$$ indeed we have $$T_{n+1}-T_n=3^{2n+1}+2^{2n+1}-3^{2n-1}-2^{2n-1}=3^{2n-1}\cdot 8+2^{2n-1}\cdot 3=3^{2n-1}(10-2)+2^{2n-1}(5-2)=5(2\cdot3^{2n-1}+2^{2n-1})-2(3^{2n-1}+2^{2n-1})$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Number of ways to put balls into bins How many ways can I put $12$ balls into $11$ bins, such that no bin contains more than $2$ balls?
The problem I am having is assigning balls to bins.
| Here is my answer, going with the assumption that having identical bins means that having two in the first and one in each other is the same as having two in the last and one in each other.
Essentially, we are figuring out different ways to add up to $12$ using eleven numbers, were the numbers being added are various ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Compute : $\int\frac{x+2}{\sqrt{x^2+5x}+6}~dx$ Question: Compute this integral
$$\int\frac{x+2}{\sqrt{x^2+5x}+6}~dx$$
My Approach:
$$\int\frac{x+2}{\sqrt{x^2+5x}+6}~dx$$
$$=\int\frac{x+2}{\sqrt{x^2+5x}+6}\times \frac{{\sqrt{x^2+5x}-6}}{{\sqrt{x^2+5x}-6}}~dx$$
$$\int\frac{(x+2)(\sqrt{x^2+5x})}{x^2+5x-36}~dx~~- \underb... | $$ \int \frac{\sqrt{x^2+5x}}{x^2+5x-36}~dx$$
$$=\int \frac{\sqrt{x^2+5x}+6-6}{(\sqrt{x^2+5x})^2-6^2}~dx$$
$=\int \frac{\sqrt{x^2+5x}+6}{(\sqrt{x^2+5x}+6)(\sqrt{x^2+5x}-6)} - \int \frac{6}{x^2+5x-36} ~dx$
$$=\int \frac{1}{\sqrt{x^2+5x}-6} - \int \frac{6}{x^2+5x-36} ~dx$$
Hope you can proceed further.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
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Find the value of $\tan^6(\frac{\pi}{7}) + \tan^6(\frac{2\pi}{7}) + \tan^6(\frac{3\pi}{7})$ I wish to find the value of $\tan^6(\frac{\pi}{7}) + \tan^6(\frac{2\pi}{7}) + \tan^6(\frac{3\pi}{7})$ as part of a larger problem. I can see that the solution will involve De Moivre's theorem somehow, but I cannot see how to ap... | Note that $\tan^2 \frac\pi7$, $\tan^2 \frac{2\pi}7$, and $\tan^2 \frac{3\pi}{7}$ are the roots of the polynomial equation $$x^3 - 21x^2 + 35x - 7 = 0.$$
If we label these roots $r$, $s$, and $t$, then Vieta's formulae tell us that $$\begin{cases}r + s + t = 21,\\ rs + rt + st = 35,\\ rst = 7.\end{cases}$$
From these, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Number Theory proof involving Legendre symbol Suppose $a,b$ are integers s.t. $ab\equiv 1\pmod{p}$. Show $(\frac{a^2+a}{p})=(\frac{b+1}{p})$.
Not really sure on how to approach this proof.
If I take $a^2+a\equiv 0\pmod{p}$
$a\equiv -a^2\pmod{p}$
$b^2a\equiv -b^2a^2\pmod{p}$
$b\equiv -1\pmod{p}$ so $b+1\equiv 0\pmod{p}$... | Since $(b^2|p) = (b|p)^2 = 1$,
$$\left(\frac{a^2 + a}{p}\right) = \left(\frac{a^2 + a}{p}\right)\left(\frac{b^2}{p}\right)= \left(\frac{a^2b^2 + ab^2}{p}\right) =\cdots$$
Complete the argument using the condition $ab \equiv 1\pmod{p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine the acceleration given the position function of a particle
The position function of a particle in a test laboratory is $s(t) = \frac{10t}
{t^2+3}$.
Determine the acceleration of the particle of particle after 4 seconds.
The speed $v(t)$ of the particle is the derivative of the position $\frac{ds(t)}{dt}$.... | Let $S(t)$ be given by
$$s(t)=\frac{10t}{t^2+3}\tag1$$
Differentiating $(1)$ yields
$$\begin{align}
v(t)&=s'(t)\\\\
&=-10\frac{t^2-3}{(t^2+3)^2}\\\\
&=-10\frac{t^2+3-6}{(t^2+3)^2}\\\\
&=-10\frac{1}{t^2+3}+60\frac{1}{(t^2+3)^2}\tag 2
\end{align}$$
Differentiating $(2)$ yields
$$\begin{align}
a(t)&=s''(t)\\\\
&=20\frac{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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If $\dfrac {1}{a+b} +\dfrac {1}{b+c}=\dfrac {3}{a+b+c}$, prove that $\angle B=60^\circ $ If $\dfrac {1}{a+b}+\dfrac {1}{b+c}=\dfrac {3}{a+b+c}$, prove that $\angle B=60^\circ$.
My Attempt
$$\dfrac {1}{a+b}+\dfrac {1}{b+c}=\dfrac {3}{a+b+c}$$
$$\dfrac {a+2b+c}{(a+b)(b+c)}=\dfrac {3}{a+b+c}$$
$$a^2-ac-b^2+c^2=0$$.
How t... | In a triangle,
$b^2=c^2+a^2−2ca \cos B$ .....(1)
From question,
$(a+2b+c)(a+b+c) = 3(a+b)(b+c)$
$a^2+ab+ac+2ab+2b^2+2bc+ca+cb+c^2=3ab+3ac+3b^2+3bc$
$a^2+c^2-ca=3b^2−2b^2$
$a^2+b^2-ca=b^2$ .....(2)
Comparing equation (1) and (2),
$a^2+b^2−2ca\cos B=a^2+b^2–ca$
$\cos B = \frac 12$
$B = 60°$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Uniform convergence of $\sum_{n=1}^{\infty}\frac{x^n}{1+x^n}$
Where does the seriers $\displaystyle \sum_{n=1}^{\infty}\frac{x^n}{1+x^n}$ converges ? On which interval does it converge uniformly ?
We have $$\lim_n \frac{x^n}{1+x^n}=\begin{cases}0 &\text{ , if } |x|<1\\\frac{1}{2} &\text{ , if } x=1\\\text{does not ex... | Observe for all $|x| \leq R<1$ you have that
\begin{align}
\left|\sum^m_{n=k} \frac{x^n}{1+x^n}\right| \leq \sum^m_{n=k} \frac{|x|^n}{1-R^n} \leq \frac{1}{1-R}\sum^m_{n=k}R^n = \frac{R^k-R^{m+k+1}}{(1-R)^2}
\end{align}
which means the series is uniformly Cauchy on $[-R, R]$. Hence the series is uniformly convergent on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Isometry and its inverse I got this affine map:
$$
f: R^3 \rightarrow R^3: \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} \rightarrow A \cdot \begin{pmatrix}x\\
y\\
z\\\end{pmatrix} + \begin{pmatrix}0\\
-1\\
1\\\end{pmatrix}
$$
with $$ A = \begin{bmatrix}
1 & a_{12} & a_{22}\\
0 & 1 & a_{21}\\
0 & 0 & 1
\end{bmatrix}$$
Also ... | $\overline{b} = g(0) = f^{-1}(0)$, so solve $f(x) = 0$ so $Ax = (0,1,-1)$. The answer (which depends on the 3 constants in $A$) is the required $\overline{b}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \le \frac{n}{n+1}$? I have a series $$a_n = \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \quad n \ge 1$$
For example, $a_3 = \frac{1}{4}+\frac{1}{5}+\frac{1}{6}$.
I need to prove that for $n \ge 1$:
$$a_n = \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \l... | $$\frac{1}{n+1} \leq \frac{1}{n+1}$$
$$\frac{1}{n+2}< \frac{1}{n+1}$$
$$\cdots$$
$$\frac{1}{2n} < \frac{1}{n+1}$$
Now add up!
Equality comes when $n=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $y=\sin{(m\cos^{-1}\sqrt{x})}$ then prove that $\lim\limits_{x\to 0}\frac{y_{n+1}}{y_n}=\frac{4n^2-m^2}{4n+2}$ If $y=\sin{(m\cos^{-1}\sqrt{x})}$ then prove that $$\lim\limits_{x\to 0}\frac{y_{n+1}}{y_n}=\frac{4n^2-m^2}{4n+2}$$
Note: $y_n = D^ny(x)$. To use: Libnitz rule of successive Differentiation.
Attempt:
$y_1=... | We have $y= \sin(m \cos ^{-1} \sqrt{x})$ & using the notation $y_n=\frac{d^n y}{dx^n}$. You start fine ... differentiating (multiple use of the chain rule) squaring & using Pythagorus to get to
\begin{eqnarray*}
4x(1-x)y_1^{2}=m^2(1-y^2)
\end{eqnarray*}
Now differentiate this
\begin{eqnarray*}
8x(1-x)y_1 y_2 +(4-8x)y_1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Find $\displaystyle\lim_{x\to\infty}x - \sqrt{x+1}\sqrt{x+2}$ using squeeze theorem
Find $$\lim_{x\to\infty}x - \sqrt{x+1}\sqrt{x+2}$$ using squeeze theorem
Tried using binomial expansion, but have no idea on how to continue.
| $$\begin{align*}&x-\sqrt{x+1}\sqrt{x+2}=\frac{x^2-(x+1)(x+2)}{x+\sqrt{x+1}\sqrt{x+2}}=\frac{-3x+2}{x+\sqrt{x^2+3x+2}}=\\{}\\
&\frac{-3+\frac2x}{1+\sqrt{1+\frac3x+\frac2{x^2}}}\xrightarrow[x\to\infty]{}\frac{-3+0}{1+\sqrt{1+0+0}}=-\frac32\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluation of the series $\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}$. Find the following sum
$$\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}$$
Could someone give some hint to proceed in this question?
| Consider that $$ \frac{(n+2)^2}{2^n} - \frac{(n+3)^2}{2^{n+1}} = \frac{n^2+2n-1}{2^n} = \frac{n^2+2n+3}{2^n} - \frac{4}{2^n}.$$
So, if the required sum is denoted by $S$, we have
$$ \sum_{n=1}^{\infty} \frac{(n+2)^2}{2^n} - \frac{(n+3)^2}{2^{n+1}} = S - 4 \sum_{n=1}^{\infty} \frac{1}{2^n}.$$
The left hand side is a te... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If the roots of $9x^2-2x+7=0$ are $2$ more than the roots of $ax^2+bx+c=0$, then value of $4a-2b+c=$?
If the roots of $9x^2-2x+7=0$ are $2$ more than the roots of $ax^2+bx+c=0$, then value of $4a-2b+c=$?
My approach is: Let the roots of $ax^2+bx+c=0$ are $\alpha$ and $\beta$. So, $$\alpha+\beta=-\frac{b}{a}\\ \alpha\... | Since $ax^2+bx+c=0$ has the same roots as $x^2+\frac{b}{a}x+\frac{c}{a}=0$ we can notice that if $(a_1,b_1,c_1)$ satisfies the requirements then also does $(ta_1,tb_1,tb_2)$ since $4a-2b+c\neq 0$ we can get $4a-2b+c=\frac{7t}{9}$ for any $t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve $\frac{2}{\sin x \cos x}=1+3\tan x$ Solve this trigonometric equation given that $0\leq x\leq180$
$\frac{2}{\sin x \cos x}=1+3\tan x$
My attempt,
I've tried by changing to $\frac{4}{\sin 2x}=1+3\tan x$, but it gets complicated and I'm stuck. Hope someone can help me out.
| Move everything to the left hand side and substitute $u=-3\tan x$ to get
\begin{align*}0&=\frac{2}{\sin x\cos x}-1-3\tan x\\ &=2\left(\tan x+\frac 1{\tan x}\right)-1-3\tan x\\ &=-1-\frac{6}{-3\tan x}-\frac{3\tan x}{3}\\ &=\frac u3 -1-\frac 6u \\ &= \frac{(u-6)(u+3)}{3u}\end{align*}
Thus we get that $u=-3$ or $u=6$. The... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate the determinant of $A-2A^{-1}$ given the characteristic and minimal polynomials of $A$ Given the characteristic polynomial and minimal polynomial of $A$ being $(x+2)^{6}(x-1)^{3}$ and $(x+2)^{2}(x-1)^{3}$, respectively. How do I determine the characteristic polynomial and minimal polynomial of $A-2A^{-1}$.
I... | $A$ is similar to a triangular matrix with a diagonal of $-2$ repeated $6$ times and $1$ repeated $3$ times. Therefore, $\det A= (-2)^6 \cdot 1^3 = 2^6$. For simplicity, let's assume $A$ is triangular.
We have $\det (A-2A^{-1}I) \det A = \det (A^2-2I)$.
The diagonal of $A^2$ is $(-2)^2$ repeated $6$ times and $1^2$ rep... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $s(N(10^{n}-1)) = s(10^{n}-1)$
Let $N$ be a positive integer, and let $n$ be the number of digits in $N$ in decimal representation. Also let $s(n)$ denote the sum of the digits of $n$. Prove that $s(N(10^{n}-1)) = s(10^{n}-1)$.
For example, we have $43 \cdot 99 = 4257$ and $s(4257) = 18 = s(99)$. How do w... | Here I am assuming that the last digit $a_0 \neq 0$, if that's not the case then go for the last non-zero digit in your number $N$ and modify the proof accordingly.
Let
$$N=a_{k-1}10^{k-1}+\dotsb +a_0, \qquad \text{ where } a_i \in \{0,1,\ldots ,9\} \text{ and } a_0 \neq 0$$
Then
\begin{align*}
N(10^k-1) & =[a_{k-1}10... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can the definite integral $ I(n) := \frac{2n}{\pi}\int_0^{2\sqrt{2n-1}} \frac{\ln(x)\sqrt{-x^2 + 8n-4}}{4n^2-x^2}dx$ be computed? For $n \in \mathbb N$, $n \geq 1$, consider the following integral expression:
\begin{equation} I(n) := \frac{2n}{\pi}\int_0^{2\sqrt{2n-1}} \frac{\ln(x)\sqrt{-x^2 + 8n-4}}{4n^2-x^2} dx \end... | Hint. By making the change of variable,
$$
x=2\sqrt{2n-1}\:u ,\quad dx=2\sqrt{2n-1}\:du,\quad u=\frac{x}{2\sqrt{2n-1}},
$$ one has
$$
\frac{\pi}{2n}\cdot I(n)=\ln(2\sqrt{2n-1})\int_0^1\frac{\sqrt{1-u^2}}{a^2-u^2}\:du+\int_0^1\frac{\ln u \cdot\sqrt{1-u^2}}{a^2-u^2}\:du \tag1
$$ with
$$
a:=\frac{n}{\sqrt{2n-1}}>1, \quad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2215805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Finding Trigonometric Fourier Series of a piecewise function Find the Fourier Trigonometric series for:
$$f(x)=
\begin{cases}
\sin(x) & 0\leq x \leq \pi
\\ 0 & \pi\leq x \leq 2\pi,
\\
\end{cases}\quad
f(x+2\pi)=f(x).$$
I tried to find the series of this function, but when I plot up to 50 terms with Wolfr... | \begin{align}
a_0 &= \frac{1}{2\pi}\int_{0}^{2\pi} \sin x \cdot \mathbf{I}_{[0,\pi]}\, dx = \frac{1}{2\pi}\int_{0}^{\pi} \sin x \, dx
= \frac{1}{\pi}\\
a_n &= \frac{2}{2\pi}\int_{0}^{2\pi} \sin x \cdot \mathbf{I}_{[0,\pi]} \cdot \cos\tfrac{2\pi n x}{2\pi}\, dx
= \frac{1}{\pi} \int_{0}^{\pi} \sin x \cdot \cos nx \, dx
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2216875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Implicit Differentiation Quesrion $$x^y = y^x$$ is the equation
My solution is :$$\frac { dy }{ dx } =\left( \ln { y-\frac { y }{ x } } \right) /\left( \ln { x-\frac { x }{ y } } \right) $$
Just wondering if this is correct!
| Your solution does not match mine:
$$
x^y = y^x \implies x\ln y = y\ln x \implies \frac{\ln y}{y} = \frac{\ln x}{x} \\
\frac{\frac1y\cdot y-1\cdot\ln y }{y^2}dy = \frac{\frac1x\cdot x-1\cdot\ln x }{x^2}dx\\
\frac{1-\ln y}{y^2} dy= \frac{1-\ln x}{x^2}dx\\
\frac{dy}{dx} = \frac{1-\ln x}{1-\ln y}\cdot\frac{y^2}{x^2}
$$
A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2218294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Integral $\int {t+ 1\over t^2 + t - 1}dt$
Find : $$\int {t+ 1\over t^2 + t - 1}dt$$
Let $-w, -w_2$ be the roots of $t^2 + t - 1$.
$${A \over t + w} + {B \over t+ w_2} = {t+ 1\over t^2 + t - 1}$$
I got $$A = {w - 1\over w - w_2} \qquad B = {1- w_2\over w - w_2}$$
$$\int {t+ 1\over t^2 + t - 1}dt = A\int {1\over t+ w} ... | Here's an alternate method: we have $$I = \int \frac{t+1}{t^2+t-1}dt$$
Note that $\frac{d}{dt}(t^2+t-1) = 2t+1$
Then write $$2I = \int \frac{2t+1}{t^2+t-1} +\frac{1}{t^2+t-1}dt$$
$$2I = \ln|t^2+t-1|+\underbrace{\int \frac{1}{(t+\frac{1}{2})^2-\frac{5}{4}} dt}_{J}$$
Note that $$\frac{1}{(t+\frac{1}{2})^2-\frac{5}{4}}= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2222679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Direct proof that $\frac{1}{\sqrt{1-x}} = \frac{1}{2\pi}\int_0^{2\pi} \frac{d\theta}{1-x\cos^2\theta}$? In a totally indirect way, I've proven to myself that
$$\frac{1}{\sqrt{1-x}} = \frac{1}{2\pi}\int_0^{2\pi} \frac{d\theta}{1-x\cos^2\theta}.$$
The proof was by expanding the integrand as a power series in $x\cos^2\the... | Here's the contour integral solution:
$$
\cos\theta=\frac{z+1/z}{2},
$$
where $z$ is belongs to the unit circle $C_1=\{z\in\mathbb C:\ z=e^{i\theta}\}$
and $d\theta=-i\,dz/z$ so
$$
\int_0^{2\pi}\frac{d\theta}{1-x\cos^2\theta}=\oint_{C_1}\frac{i4zdz}{xz^4+2(x-2)z^2+x}.
$$
If $x=0$, we get $-i\int_{C_1}dz/z=2\pi$. Otherw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2225351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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mathematical induction with exponent $$\frac 13 + \frac 1{3^2} + \frac 1{3^3} + \dots + \frac 1{3^n} + = \frac 12 \times \left( 1 - \frac{1}{3^n} \right)$$
Step 1 - $n=1$
$$\begin{align}
\frac 1 {3^1} & = \frac 1 2 \times \left( 1 - \frac 1 {3^1} \right) \\
\frac 1 3 & = \frac 1 2 \times \left( 1 - \frac 1 3 ... | By hypothesis, we know that
$$
\frac{1}{3}+\frac{1}{3^2}+\dots+\frac{1}{3^k}=\frac{1}{2}\bigg[1-\frac{1}{3^k}\bigg]
$$
Hence, by adding to both sides by $\frac{1}{3^{k+1}}$ we get
$$\begin{align}
\frac{1}{3}+\frac{1}{3^2}+\dots+\frac{1}{3^k}+\frac{1}{3^{k+1}}&=\frac{1}{2}\bigg[1-\frac{1}{3^k}\bigg]+\frac{1}{3^{k+1}}\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Is it true that $f$ must be constant? Let $f: \Bbb R \to \Bbb R$ be a function which satisfies $f(x) = f(2x-1)$ for all $x \in \Bbb R$ and $f$ is continuous at $x=1$. Is it true that $f$ must be constant?
| From your assumption we have $f(x)=f(\frac{x}{2}+\frac{1}{2})$ for all $x$. For $x\in \mathbb R$ and substituting successively $x$ by $\frac{x}{2}+\frac{1}{2}$, we get
$$f(x)=f\Big(\frac{x}{2}+\frac{1}{2}\Big)=f\Big(\frac{1}{2}\Big(\frac{x}{2}+\frac{1}{2}\Big)+\frac{1}{2}\Big)=f\Big(\frac{x}{2^2}+\frac{1}{2}+\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Find the value of $(\log_{a}b + 1)(\log_{b}c + 1)(\log_{c}a+1)$ if $\log_{b}a+\log_{c}b+\log_{a}c=13$ and $\log_{a}b+\log_{b}c+\log_{c}a=8$ Let $a,b$, and $c$ be positive real numbers such that
$$\log_{a}b + \log_{b}c + \log_{c}a = 8$$
and
$$\log_{b}a + \log_{c}b + \log_{a}c = 13.$$
What is the value of
$$(\log_{a}b... | Notice that
$$
\log_uv\log_vw=\dfrac{\log v}{\log u}\cdot\dfrac{\log w}{\log v}=\dfrac{\log w}{\log u}=\log_uw \quad \forall u,v,w>0
$$
therefore
\begin{eqnarray}
(\log_ab+1)(\log_bc+1)(\log_ca+1)&=&(\log_ab\log_bc+\log_ab+\log_bc+1)(\log_ca+1)\\
&=&(\log_ac+\log_ab+\log_bc+1)(\log_ca+1)\\
&=&\log_ac\log_ca+\log_ac+\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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"answer_id": 1
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How to do junior math Olympiad question Let $a$ and $b$ be such that $0<a<b$. Suppose that $a^3 = 3a-1$ and $b^3 = 3b-1$. Find $b^2 -a$.
| Since $0<a<b$ and $a^3=3a-1$ and $b^3=3b-1$, then we need to find the two positive real roots of $x^3-3x+1=0$ with $a$ being the smaller root and $b$ being the bigger root.
Following Cardano's solution, if we let $x=u+v$ then
$$
\begin{align*}
x^3-3x+1&=0\\
(u+v)^3-3(u+v)+1&=0\\
u^3+3u^2v+3uv^2+v^3-3u-3v+1&=0\\
u^3+v^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Finding the exact value of a radical How do I show that $\sqrt{97 +56\sqrt3}$ reduces to $7 +4\sqrt3?$. Without knowing intitially that it reduces to that value.
| While the accepted solution is fine if you're willing to solve a quadratic equation, it's worth developing some theory to avoid this for two reasons:
*
*To save effort spent solving quadratic equations, replacing it by effort spent doing these different and exciting calculations instead :)
*So that we can also figu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find the value of $\sin (x)$ if $\cos(x)=\frac 1 3$ $$\cos (x)=\left (\frac{1}{3}\right)$$for $0°\lt\ x\lt90°$
Find the exact value of $\sin(x)$.
Give your answer as a surd.
I found the value with my calculator
$$\sin (x)=\left(\frac{\sqrt 8}{3}\right)$$
Is there a way of doing this by hand?
| $\sin^2 x + \cos^2 x = 1$
$\sin^2 x = 1 - \cos^2 x$
$\sin x = \pm \sqrt{1 - \cos^2 x}$
$\sin x = \pm \sqrt{1 - \left( \frac 13\right)^2}$
$\sin x = \pm \sqrt{1 - \frac 19}$
$\sin x = \pm \sqrt{\frac 89}$
$\sin x = \pm \frac {\sqrt 8}3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determinant doubt How do we prove, without actually expanding, that
$$\begin{vmatrix}
\sin^2 {A}& \cot {A}& 1\\
\sin^2 {B}& \cot{B}& 1\\
\sin^2 {C}& \cot{C}& 1
\end{vmatrix}=0$$
where $A,B,C$ are angles of a triangle?
I tried applying cosine double angle formula but couldn't get anywhere.
| It is enough to show that
$$ \det\begin{pmatrix}\sin^3 A & \cos A & \sin A \\ \sin^3 B & \cos B & \sin B \\ \sin^3 C & \cos C & \sin C \end{pmatrix}=0 $$
or that
$$ \det\begin{pmatrix}a^3 & \frac{b^2+c^2-a^2}{bc} & a \\ b^3 & \frac{a^2+c^2-b^2}{ac} & b \\ c^3 & \frac{a^2+b^2-c^2}{ab} & c \end{pmatrix}=0 $$
or that
$$ \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simple inequality $\left|\frac{3x+1}{x-2}\right|<1$
$$\left|\frac{3x+1}{x-2}\right|<1$$
$$-1<\frac{3x+1}{x-2}<1$$
$$-1-\frac{1}{x-2}<\frac{3x}{x-2}<1-\frac{1}{x-2}$$
$$\frac{-x+1}{x-2}<\frac{3x}{x-2}<\frac{x-3}{x-2}$$
$$\frac{-x+1}{x-2}<\frac{3x}{x-2}<\frac{x-3}{x-2} \text{ , }x \neq 2$$
$${-x+1}<{3x}<{x-3} \text{ , ... | Multiplying by $x-2$ changes sign of the inequality when $x-2<0$ though you could multiply by something which is always $>0$ for example multiplying by $(x-2)^2$(assuming $x\neq2$) we get
$$\frac{-x+1}{x-2}<\frac{3x}{x-2}<\frac{x-3}{x-2}\\(1-x)(x-2)<3x(x-2)<(x-2)(x-3)$$
Which is equivalent with
$$(4x-1)(x-2)>0\land (x-... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 6
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Cyclicity:To find the units digit of a number
Can you please help me to find the units digit of this number:
33^34^35^36^37^38 .. I got this question in CSR 2017
| $33^4\equiv 3^4\equiv 81\equiv 1\pmod{10}$.
$34^{35^{36^{37^{38}}}}=4k$ for some $k\in\mathbb Z^+$.
$33^{4k}\equiv \left(33^4\right)^k\equiv 1^k\equiv 1\pmod{10}$.
Euler's theorem is relevant here, but I've been able to explain this simply without it.
Edit: The multiplicative order of $33$ modulo $10$ is $4$.
You coul... | {
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"timestamp": "2023-03-29T00:00:00",
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Simple inequality $\frac{4x+1}{2x-3}>2$
$$\frac{4x+1}{2x-3}>2$$
I have started with looking at positive/negative situations and there are 2 or that both expressions are positive or both are negative.
If both a positive we can solve for $$\frac{4x+1}{2x-3}>2$$
$${4x+1}>2(2x-3)$$
$${4x+1}>4x-6$$
$${0x}>-7$$
Or both a... | For $x > \frac{3}{2}$, we rearrange to $$4x+1 > 4x -6 \iff 1 > -6$$ and so it holds for all $x > \frac{3}{2}$.
If $x < \frac{3}{2}$, we rearrange to $$4x+1 < 4x-6 \iff 1 < -6$$ and hence it does not hold for $x < \frac{3}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2242281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Which point of the sphere $x^2+y^2+z^2=19$ maximize $2x+3y+5z$? Which point of the sphere $x^2+y^2+z^2=19$ maximize $2x+3y+5z$?
So I assume that there is a point maximizing $2x+3y+5z$. How can I calculate the exact value of this point?
| By the Cauchy-Schwarz inequality
$$ 2x+3y+5z \leq \sqrt{x^2+y^2+z^2} \sqrt{2^2+3^2+5^2} = 19 \sqrt{38} $$
and equality is achieved if $(x,y,z)=\lambda(2,3,5)$, i.e. for $\lambda=\frac{1}{\sqrt{2}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2246736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
I spent a lot of time trying to solve this and, having consulted some ... | $x^2+y^2+z^2$ only depends on the squared distance of $(x,y,z)$ from the origin and the constraint $x+y+z=1$ tells us that $(x,y,z)$ lies in a affine plane. The problem is solved by finding the distance between such plane and the origin: since the plane is orthogonal to the line $x=y=z$,
$$ \min_{x+y+z=1}x^2+y^2+z^2 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 15,
"answer_id": 4
} |
In a triangle $ABC$, if its angles are such that $A=2B=3C$ then find $\angle C$? In a triangle $ABC$, if its angles are such that $A=2B=3C$ then find $\angle C$?
$1$. $18°$
$2$. $54°$
$3$. $60°$
$4$. $30°$
My Attempt:
$$A=2B=3C$$
Let $\angle A=x$ then $\angle B=\dfrac {x}{2}$ and $\angle C=\dfrac {x}{3}$
Now,
$$x+\dfr... | As A=2B=3C,
B=1.5C
Therefore, summing the angles, A+B+C= 3C+1.5C+C=180 degrees.
5.5C=180
C=36/1.1
=360/11
= 32.77 degrees.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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How do I prove using the definition of limit that $\lim_{(x,y)\rightarrow (0,0)}\frac {x^2+y^3}{y^2 - x + xy}$? I want to use the definition of limit in order to prove $\lim_{(x,y)\rightarrow (0,0)}\frac {x^2+y^3}{y^2 - x + xy}$, but I almost crack my head open trying to do this, it seems to be easy way but y cant find... | Note that
$$
x=y^2\implies\frac{x^2+y^3}{y^2-x+xy}=\frac{y^4+y^3}{y^2-y^2+y^3}=y+1\overset{y\to0}\to1
$$
and that
$$
x=0\implies\frac{x^2+y^3}{y^2-x+xy}=\frac{y^3}{y^2}=y\overset{y\to0}{\to}0
$$
These show that the limit does not exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2248758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Given: $\int f(x)\sin x \cos x dx = \log(f(x)){1\over 2( b^2 - a^2)}+C$, find $f(x)$.
$$\int f(x)\sin x \cos x dx = \log(f(x)){1\over 2( b^2 - a^2)}+C$$
On differentiating, I get,
$$f(x)\sin x\cos x = {f^\prime(x)\over f(x)}{1\over 2( b^2 - a^2)}$$
$$\sin 2x (b^2 - a^2) = {f^\prime( x)\over (f(x))^2} $$
On integr... | The solutions are OK (contrary to what I first wrote). After integrating, you can check that you have
$$
\frac{1}{f}=\frac{(b^2-a^2)\cos 2x}{2}
$$
and, according to the solution in the book,
$$
\frac{1}{f}=a^2\sin^2x+b^2\cos^2x.
$$
The difference should be a constant (the integrating constant). And indeed,
$$
\frac{(b^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
show that $a^2-1$ is a quadratic nonresidue mod $p$ if and only if $p\equiv 6\pmod 7$ Question: Let prime number $p>7$, for positive integer $a,b,c$ such $1<a<b<c<p$, and
$$a+b+c\equiv a^3+b^3+c^3\equiv a^5+b^5+c^5\equiv\frac{p-1}{2}\pmod p$$
show that:
$a^2-1$ is a quadratic nonresidue mod $p$ if and only if $p\equiv... | Since $ab+bc+ac$ and $abc$ are rational fractions in $a+b+c,a^3+b^3+c^3,a^5+b^5+c^5$,
you can show that for $p \neq 2,3$, then $a,b,c$ satisfy these relations mod $p$ if and only if they are roots of a certain cubic equation.
(in fact, you get the equation $8a^3+4a^2-4a-1 = 0$)
It turns out that over $\Bbb C$, its roo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
for what values of N element N is $3^{n+1}>n^4$ So I'm doing homework for my logic class...
For what values of n element of natural numbers is $3^{n+1}>n^4$ and we are supposed to use induction. This is what I have so far...
Proof: The inequality holds for n=1. Since $3^2>1^4= 9>1$. Assume $3^{k+1}>k$, where $k\geq 1... | The inequality $3^{n+1}>n^4$ is true for $n=1,\ 2$ and all $n\ge5$. (The inequality is false for $n=3$ and $n=4$ because $81\not>81$ and $243\not>256$.)
We can prove the inequality for $n\ge5$ by induction.
The base case ($n=5$) is true: $3^6>5^4, \ \ 729>625$.
Now suppose the inequality is true for some $k\ge5$:
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Probability of drawing a spade on the first draw, a heart on the second draw, and an ace on the third draw My answer is
$(\frac{1}{52})(\frac{1}{51})(\frac{2}{50})$ + $2(\frac{1}{52})(\frac{12}{51})(\frac{3}{50})$ + $(\frac{12}{52})(\frac{12}{51})(\frac{4}{50})$
= $\frac{1}{204}$
But the answer in the textbook is
$(\fr... | The probability of drawing a spade followed by a heart followed by the ace of spades or hearts is
$$\frac{12\times13}{51\times 50}\times\frac2{52}.$$
The probability of drawing a spade followed by a heart followed by the ace of diamonds or clubs is is
$$\frac{13^2}{51\times 50}\times\frac2{52}.$$
These add to
$$\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which is greater $x_1$ or $x_2$? $$x_1=\arccos\left(\frac{3}{5}\right)+\arccos\left(\frac{2\sqrt{2}}{3}\right)$$
$$x_2=\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{2\sqrt{2}}{3}\right)$$ We have to find which is greater among $x_1$ and $x_2$
If we add both we get $$x_1+x_2=\pi$$ If we use formulas we get
$$x_1=\a... | We have
$$\frac{3}{5} > \frac{1}{2} \implies \arcsin \frac{3}{5} > \arcsin \frac{1}{2} = \frac{\pi}{6}$$
and
$$\frac{2\sqrt{2}}{3} > \frac{\sqrt{3}}{2} \implies \arcsin \frac{2\sqrt{2}}{3} > \arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3},$$
hence
$$x_2 > \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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$10^n+1$ as a product of consecutive primes I am interested to know the examples or the systematic rule whether $10^n+1$ is a product of consecutive primes.
For example,
when $n=1$,
we have $10^1+1=11$ is a product of a single prime 11. So, a True example.
When $n=2$,
we have $10^2+1=101$ is a product of a single prim... | I'm finding that $11$ divides $10^n+1$ for all odd $n$. $7*11*13$ divides $10^n+1$ for odd $n$ divisible by $3$. But $17$ is a divisor only if $n$ is an odd multiple of $8$. Thus it seems $10^n+1$ is not a product of consecutive primes for odd $n>3$. If even $n$ is an odd multiple of $2$, $101$ divides $10^n+1$, but $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding a sum of $1+\frac{1}{4\cdot2^{4}}+\frac{1}{7\cdot2^{7}}+\frac{1}{10\cdot2^{10}}+\cdots$ I need someone to find a mistake in my soliution or maybe to solf it much more easily... I have got a sum $$1+\frac{1}{4\cdot2^{4}}+\frac{1}{7\cdot2^{7}}+\frac{1}{10\cdot2^{10}}+\cdots$$ and need to evaluate it. So here's my... | Sometimes when faced by one of these "sections" of a well-known series it helps to think "roots of unity". The other solutions given here are surely the way to go in this case, but it's worth knowing the generic technique, so here goes.
Let $G(x)=\sum_0^\infty \frac{x^{n+1}}{n+1}$ we see that we want "one-third" of thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Vectors representing the principal axes of an ellipse The principal axes of the ellipse given by the equation $ 9x^2+6y^2-4xy=4$ are along the directions of the vectors
a) $2i+3j$ and $3i-2j$
b) $2i+j$ and $i-2j$
c) $3i+2j$ and $2i-3j$
d) $i+2j$ and $2i-j$
e) $i+j$ and $i-j$
Seems trivial however I do not have much ide... | Notice that
$$ 9x^2+6y^2-4xy = \begin{pmatrix} x& y \end{pmatrix} \begin{pmatrix} 9 & \frac{-4}{2} \\ \frac{-4}{2} & 6 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$
We'll define $\vec{x} = \begin{pmatrix} x \\ y \end{pmatrix} $. Now:
$$ 9x^2+6y^2-4xy = \vec{x}^T \begin{pmatrix} 9&-2\\ -2&6 \end{pmatrix} \vec{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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If $a,b,c$ are positive, prove that $\frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a} \geq \frac{9}{a+b+c}$ If $a,b,c$ are positive real numbers, prove that
$$\frac{2}{a+b}+\frac{2}{b+c}+ \frac{2}{c+a}≥ \frac{9}{a+b+c}$$
| your inequality is equivalent to
$$2\,{a}^{3}-{a}^{2}b-{a}^{2}c-a{b}^{2}-a{c}^{2}+2\,{b}^{3}-{b}^{2}c-b{c
}^{2}+2\,{c}^{3}>0
>0$$ after Clearing the denominators and this is equivalent to
$$(a-b)(a^2-b^2)+(a-c)(a^2-c^2)+(b-c)(b^2-c^2)\geq 0$$ which is true.
the equal sign holds if $$a=b=c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If $a, b, c$ are positive and $a+b+c=1$, prove that $8abc\le\ (1-a)(1-b)(1-c)\le\frac{8}{27}$ If $a, b, c$ are positive and $a+b+c=1$, prove that $$8abc\le\ (1-a)(1-b)(1-c)\le\dfrac{8}{27}$$
I have solved $8abc\le\ (1-a)(1-b)(1-c)$ (by expanding $(1-a)(1-b)(1-c)$)
but do not get how to show that $(1-a)(1-b)(1-c)\le\fra... | \begin{eqnarray*}
a(b-c)^2+b(c-a)^2+c(a-b)^2 \geq 0
\end{eqnarray*}
This can be rearranged to
\begin{eqnarray*}
9abc \leq (ab+bc+ca)(a+b+c) \\
8abc \leq 1-(a+b+c)+(ab+bc+ca)-abc
\end{eqnarray*}
Thus the first inequality is shown.
\begin{eqnarray*}
4((a+b)(a-b)^2+(b+c)(b-c)^2+(c+a)(c-a)^2)+a(b-c)^2+b(c-a)^2+c(a-b)^2 \ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2264398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding $\lim_{n \to \infty} x_n$ Let $x$ be a positive real number. A sequence $x_n$ of real numbers is defined as follows:
$$x_1=\frac 12\left(x+\frac5x\right)$$ and $$x_{n+1}=\frac 12\left(x_n+\frac 5{x_n}\right) \quad\forall n\geq 1$$
The question is to show that $$\frac{x_n-\sqrt 5}{x_n+\sqrt 5}=\left(\frac{x-\sqr... | If it true for $n=k$ then we have $\frac{x_k - \sqrt{5}}{x_k + \sqrt{5}} = \left(\frac{x-\sqrt{5}}{x+\sqrt{5}}\right)^{2^k}$.
Then $$\frac{x_{k+1} - \sqrt{5}}{x_{k+1} + \sqrt{5}} = \frac{x_k + \frac{5}{x_k} - 2\sqrt{5}}{x_k + \frac{5}{x_k} + 2\sqrt{5}} = \frac{x_k^2 - 2x_k\sqrt{5} +5}{x_k^2 + 2x_k\sqrt{5}+5}$$
which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2265615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find maximum and minimum value of inverse function .
We have to find maximum and minimum value of $x^2 +y^2$
My try
how can I proceed
| Let $\lfloor xy \rfloor=n$, then
$$\frac{x^2+y^2}{1-x^2y^2} = \tan \frac{n\pi}{4}$$
You've checked that no solutions for $0\le xy \le 1$, hence the tangent value must be negative which is achievable when $n=3,7,\ldots, 4k-1, \ldots$
Now
\begin{align}
\frac{x^2+y^2}{1-x^2y^2} &= \tan \left( k\pi-\frac{\pi}{4} \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2265974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$a+b+c+d+e=79$ with constraints How many non-negative integer solutions are there to $a+b+c+d+e=79$ with the constraints $a\ge7$, $b\le34$ and $3\le c\le41$?
I get that for $a\ge7$ you do $79-7=72$, $\binom{72+5-1}{5-1}=\binom{76}4$. For $b\ge35$ I think it's $\binom{47}4$ and I'm not too sure what it is for $3\le c\le... | We may solve this problem by finding the number of solutions with the following constraints
(1) $a\ge7 $ and $c\ge3$
(2) $a\ge7 $ and $c\ge42$
(3) $a\ge7 $, $b\ge35$ and $c\ge3$
(4) $a\ge7 $, $b\ge35$ and $c\ge42$
The answer to the problem is the answer of (1) - the answer of (2) - the answer of (3) + the answer of (4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2267913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
How to work out value of $a$ and $b$ in exponents? I am a student and I need a step by step solution in working let the value of $a$ and $b$ in this question.
$$1800 = 2^3 × 3^a × 5^b$$
I don't know what steps to take in order to work out what the powers are.
Thank you and help is appreciated.
| Sweep through the prime numbers $2, 3, 5, 7, 9, 13, \dots $ as indicated by @projectilemotion.
First prime: $p=\boxed{2}$
$$
\frac{1800}{2^{3}} = 255
$$
Decomposition is now $1800 = 2^{3} \times ?$
Next prime: $p=\boxed{3}$
$$
\underbrace{\frac{225}{3} = 75}_{1}
\qquad \Rightarrow \qquad
\underbrace{\frac{75}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Question on a tricky Arithmo-Geometric Progression::
$$\dfrac{1}{4}+\dfrac{2}{8}+\dfrac{3}{16}+\dfrac{4}{32}+\dfrac{5}{64}+\cdots\infty$$
This summation was irritating me from the start,I don't know how to attempt this ,tried unsuccessful attempts though.
| Apply $(1-x)^{-2}=1+2x+3x^2+4x^3+\cdots\infty$.
Now,
$\dfrac{1}{4}+\dfrac{2}{8}+\dfrac{3}{16}+\dfrac{4}{32}+\dfrac{5}{64}+\cdots\infty\\
=\dfrac{1}{4}\left(1+\dfrac{2}{2}+\dfrac{3}{2^2}+\dfrac{4}{2^3}+\dfrac{5}{2^4}+\cdots\infty\right)\\
=\dfrac{1}{4}\left(1-\dfrac{1}{2}\right)^{-2}\hspace{25pt}\text{ here }x=\dfrac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Infinite Sum Calculation: $\sum_{k=0}^{\infty} \frac{1}{(2k+1)^2} = \frac{3}{4} \sum_{n=1}^{\infty} \frac{1}{n^2}$ Problem
Show the following equivalence: $$\sum_{k=0}^{\infty} \frac{1}{(2k+1)^2} = \frac{3}{4} \sum_{n=1}^{\infty} \frac{1}{n^2}$$
Good afternoon, dear StackExchange community. I'm studying real analysis ... | Whoa, back up a bit!
$$\begin{align}\sum_{n=1}^{\infty}\frac{1}{n^2} &= 1 + \sum_{k=1}^{\infty} \frac{1}{(2k)^2} + \sum_{k=1}^{\infty}\frac{1}{(2k+1)^2}\\&= \frac1{2(0)+1} + \sum_{k=1}^{\infty} \frac{1}{(2k)^2} + \sum_{k=1}^{\infty}\frac{1}{(2k+1)^2}\\&=\sum_{k=1}^{\infty} \frac{1}{(2k)^2} + \sum_{k=0}^{\infty}\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Differentiation - Power rule Find derivative of this function
$$g(t)=-3t(6t^4-1)^{4/3}$$
I have tried it till the answer:
$ -3t.\frac{4}{3} (6t^4 - 1)^{\frac{4}{3}-1} .\frac{d}{dt}(6t^4 -1) $
$-3t x \frac{4}{3} (6t^4-1)^\frac{1}{3} ( 6X4 t^3 -0 ) $
$ -4t (6t^4 -1)^\frac{4}{3} (24t^3) $
$-96t(6t^4-1)^\frac{1}{3} $
Ho... | We can separate the $-3$ first:
$$g'(t)=-3(t(6t^4-1)^{4/3})'$$
Then apply the product rule and chain rule:
$$=-3(t'(6t^4-1)^{4/3}+t((6t^4-1)^{4/3})')$$
$$=-3((6t^4-1)^{4/3}+4/3\cdot t(6t^4-1)^{1/3}(6t^4-1)')$$
$$=-3((6t^4-1)^{4/3}+4/3\cdot t(6t^4-1)^{1/3}(24t^3))$$
Simplify:
$$=-3((6t^4-1)(6t^4-1)^{1/3}+4/3\cdot t(6t^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2268700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Approximating arc length I have the integral expression for the arc length:
$$ \int \sqrt{1 + \frac{1}{4x}\,}\,\mathrm{d}x $$
and need to approximate the arc length of the curve: $y = $$ \sqrt{x}$
in between x = 0 and x = 4.
Should I be using the trapezoidal rule or Simpson's rule for this? And how can I begin to appl... | The formula for arc length of $y = \sqrt{x}$ from $x = 0$ to $x=4$ is
$$ \int_{0}^{4} \sqrt{ 1 + \left( \frac{dy}{dx} \right)^2 } dx$$
where here $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$, squaring this results in $\left( \frac{dy}{dx} \right)^2 = \frac{1}{4x}$. Hence our task is to integrate
$$\int_{0}^{4} \sqrt{1 + \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272482",
"timestamp": "2023-03-29T00:00:00",
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} |
If $a\cos(θ)=b$ $\cos( θ+2π/3)=c\cos(θ+4π/3)$, prove that $ab+bc+ca=0.$ THE PROBLEM: If $a\cos(θ)=b$ $\cos( θ+2π/3)=c\cos(θ+4π/3)$, prove that $ab+bc+ca=0.$
MY THOUGHT PROCESS: We have to prove that $ab+bc+ca=0$.
One method using which we can do this is, if we can somehow obtain the equation $k(ab+bc+ca)=0$ we can ded... | Using the addition formulas:
$$2\cos\left(\theta+\frac{2\pi}{3}\right)=-\cos\theta-\sqrt3\sin\theta$$
$$2\cos\left(\theta+\frac{4\pi}{3}\right)=-\cos\theta+\sqrt3\sin\theta$$
Then on the one hand
$$\cos\left(\theta+\frac{2\pi}{3}\right)\cos\left(\theta+\frac{4\pi}{3}\right)+\cos\theta\cos\left(\theta+\frac{4\pi}{3}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2273226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Prove: $\frac{a}{m_a}+\frac{b}{m_b}+\frac{c}{m_c}\ge 2\sqrt{3}$ Let $a,b,c$ be the lengths of the sides of triangle $ABC$ opposite $A,B,C$, respectively, and let $m_a,m_b,m_c$ be the lengths of the corresponding angle medians. Prove:
$$\frac{a}{m_a}+\frac{b}{m_b}+\frac{c}{m_c}\ge 2\sqrt{3}.$$
Source: I thought about it... | Another way.
By the Ptolemy and C-S twice we obtain:
$$\sum_{cyc}\frac{a}{m_a}=\sqrt{\sum_{cyc}\left(\frac{a^2}{m_a^2}+\frac{2ab}{m_am_b}\right)}\geq\sqrt{\sum_{cyc}\left(\frac{a^2}{m_a^2}+\frac{2ab}{\frac{c^2}{2}+\frac{ab}{4}}\right)}=$$
$$=2\sqrt{\sum_{cyc}\left(\frac{a^2}{2b^2+2c^2-a^2}+\frac{2ab}{ab+2c^2}\right)}=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find all solutions to $|x^2-2|x||=1$ Firstly, we have that $$\left\{
\begin{array}{rcr}
|x| & = & x, \ \text{if} \ x\geq 0 \\
|x| & = & -x, \ \text{if} \ x<0 \\
\end{array}
\right.$$
So, this means that
$$\left\{
\begin{array}{rcr}
|x^2-2x| & = & 1, \ \text{if} \ x\geq 0 \\
|x^2+2x| & = & 1, \ \tex... | $x^{2}-2x = 1 $ ,if $x^{2}\geq 2x$ and $x\geq 0$ (because$\left |x^{2}-2x \right | = 1 $ if $x\geq 0$)
but $1-\sqrt{2} \leq 0$
$x_{3} =-1+\sqrt{2}$ is not correct solution because of same reason in second equation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
find the value $\frac{49b^2+39bc+9c^2}{a^2}=52$ Let $a,b.c$ be real numbers such that
$$\begin{cases}a^2+ab+b^2=9\\
b^2+bc+c^2=52\\
c^2+ca+a^2=49
\end{cases}$$
show that
$$\dfrac{49b^2+39bc+9c^2}{a^2}=52$$
I have found this problem solution by geomtry methods.solution 1,can you someone have Algebra methods?
| Hint:
$$\begin{cases}a^2+ab+b^2=x\\
b^2+bc+c^2=y\\
c^2+ca+a^2=z
\end{cases}$$
Then
$$\begin{cases}
(a-c)(a+b+c) = x-y\\
(b-a)(a+b+c) = y-z\\
(c-b)(a+b+c) = z-x\\
\end{cases}$$
Let $(a+b+c)^{-1}=k$, then
$$\begin{cases}
(a-c) = (x-y)k\\
(b-a) = (y-z)k\\
(c-b) = (z-x)k\\
\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find $A^{50}$ for a $3\times 3$ matrix
If $$A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$$ then $A^{50}$ is:
*
*$\begin{bmatrix}1&0&0\\25&1&0\\25&0&1\end{bmatrix}$
*$\begin{bmatrix}1&0&0\\50&1&0\\50&0&1\end{bmatrix}$
*$\begin{bmatrix}1&0&0\\48&1&0\\48&0&1\end{bmatrix}$
*$\begin{bmatrix}1&0&0\\24&1&0\\24&0... | Sometimes a good way to start a problem like this is just to try a few multiplications and look for a pattern.
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$
$\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{array} \right)$$=$
$\left( \begin{array}{ccc}
1 & 0 & 0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2284631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3$
Let $x^3+ax^2+bx+c=0$ are $\alpha, \beta, \gamma$. Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3$
My attempt,
As I know from the original equation,
$\alpha+\beta+\gamma=-a$
$\alpha\beta+\beta\gamma+\alpha\gamma=b$
$\alpha\beta\gamma=-c$
I'... | Let $p_k = \alpha^k + \beta^k + \gamma^k$. By Newton identities, we have
$$\begin{align}
p_1 + a = 0 &\implies p_1 = -a\\
p_2 + ap_1 + 2b = 0&\implies p_2 = a^2 - 2b\\
p_3 + ap_2 + bp_1 + 3c = 0&\implies p_3 = -a^3 + 3ab - 3c
\end{align}
$$
In particular, we have
$$\alpha^3 + \beta^3 + \gamma^3 = p_3 = -a^3 + 3ab - 3c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2285064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Probability problem gives different answers with different methods
In a box there are $4$ red ball, $5$ blue balls and $6$ green balls.$4$ balls are picked at random. What is the probability the at least one ball is green and at least one ball red $?$
Method $1$:
Now,in how many ways can this be done $?$
$1.$ $1$ red... | Let $R$ be the number of reds and $G$ be the number of greens.
Then
\begin{align*}
P(R\geq 1, G\geq 1) &= 1-P(R=0 \cup G = 0) \\
&= 1-[P(R=0)+P(G=0)-P(R=0, G=0)]\\
&= 1 -\left[\frac{\binom{4}{0}\binom{11}{4}}{\binom{15}{4}}+\frac{\binom{6}{0}\binom{9}{4}}{\binom{15}{4}}-\frac{\binom{5}{4}\binom{10}{0}}{\binom{15}{4}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2285972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the integral $\int \frac{x-1}{(x+1) \sqrt{x^3+x^2+x}}dx$ The integral to be evaluated is $$\int \frac{(x-1)}{(x+1) \sqrt{x^3+x^2+x}}dx$$
I split the integral to obtain
$$\int \frac{\sqrt x}{(x+1) \sqrt{x^2+x+1}}dx - \int \frac{1}{(x+1)\sqrt{x^3+x^2+x}}dx$$
But I could not proceed any further, as I am not ab... | Notice
$$\begin{align}
\int \frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}dx
= &\int \frac{x-1}{(x+1)\sqrt{x+1+x^{-1}}}\frac{dx}{x}\\
= &\int \frac{x-1}{(x+1)\sqrt{x+1+x^{-1}}}\frac{d(x+1+x^{-1})}{x-x^{-1}}\\
= &\int \frac{d(x+1+x^{-1})}{(x+2+x^{-1})\sqrt{x+1+x^{-1}}}
\end{align}
$$
Let $u = \sqrt{x + 1 +x^{-1}}$, the indefinite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2286670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Matrix representation problem Consider the next linear operator:
$$T:M_{2\times2}(\mathbb{R})\longrightarrow M_{2\times2}(\mathbb{R})$$
such that $T\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=\left(\begin{array}{cc}0&a+b\\0&0\end{array}\right)$. I need find the $Im(T)$ and the $Ker(T)$ but I don't how is the matr... | We note that the vector space $M_{2\times2}(\mathbb{R})$ has dimension $4$, since we have the following basis for the vector space:
$$
\mathbf{v}_1 = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix} \qquad
\mathbf{v}_2 = \begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix} \qquad
\mathbf{v}_3 = \begin{pmatrix}0 & 0 \\ 1 & 0 \end{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
how many $3\times 3$ matrices with entries from $\{0,1,2\}$.
How many $3 × 3$ matrices $M$ with entries from $\left\{0, 1, 2\right\}$ are there for which taken from
the sum of the main diagonal of $M^TM$ is $5$.
Attempt: Let $M = \begin{pmatrix}
a & b & c\\
d & e & f\\
g & h & i
\end{pmatrix}$. where $a,b,c,d,e,f... | The numbers $a^2$ etc. are either $0$, $1$or $4$. To get them adding to $5$ you need five $1$s, or a $4$ and a $1$ and all the rest $0$s.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2291491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
Factorise $x^5+x+1$
Factorise $$x^5+x+1$$
I'm being taught that one method to factorise this expression which is $=x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1$
$=x^3(x^2+x+1)-x^2(x^2+x+1)+x^2+x+1$
=$(x^3-x^2+1)(x^2+x+1)$
My question:
Is there another method to factorise this as this solution it seems impossible to invent it?
| With algebraic identities, this is actually rather natural:
Remember if we had all powers of $x$ down to $x^0=1$, we could easily factor:
$$x^5+x^4+x^3+x^2+x+1=\frac{x^6-1}{x-1}=\frac{(x^3-1)(x^3+1)}{x-1}=(x^2+x+1)(x^3+1),$$ so we can write:
\begin{align}x^5+x+1&=(x^2+x+1)(x^3+1)-(x^4+x^3+x^2)\\
&=(x^2+x+1)(x^3+1)-x^2(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 5
} |
In Fig : what is $\angle DAC= ?$ In Fig : what is $$\angle DAC= ? $$
in triangle $\triangle ABC $ And $|AB|=|AC|$And Point $D$ in the triangle $\triangle ABC $
such that : $\angle BCD =40,\angle CBD =20$
My Try:
$$\frac{\sin 30}{AD}=\frac{\sin \angle BAD}{BD}$$
and :
$$\frac{\sin 10}{AD}=\frac{\sin \angle DAC}{DC}$$... | Let $x = \angle DAB, y = \angle DAC$. Since sum of all angles of the outer triangle is $360^\circ$, so $x + y = 80$. You can get another equation using the sine rule for the sides $BD, AD, CD$;
$$ 1 = \frac{BD}{AD} . \frac{AD}{CD} . \frac{CD}{BD} = \frac{\sin x}{\sin 30} . \frac{\sin 10}{\sin y} . \frac{\sin 20}{\sin 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find natural (arc length) parametrization of a curve. If a curve is defined like this:
$$
\rho = 1 - \cos(x)
$$
find natural parametrization of it.
So this is how the curve looks like when when I use polar coordinates $x = \rho \cos(t)$, $y = \rho\sin(t)$:
$$
\alpha(t) = \bigl((1 - \cos(t)) \cos(t), (1 - \cos(t)) \si... | The polar graph $\rho = f(\theta)$ parametrized by
$$
(x, y) = \bigl(f(t)\cos t, f(t) \sin t\bigr)
\tag{1}
$$
has speed
$$
\frac{ds}{dt} = \sqrt{f(t)^{2} + f'(t)^{2}}.
$$
For $f(t) = 1 - \cos t$, the speed is
$$
\sqrt{(1 - \cos t)^{2} + (\sin t)^{2}}
= \sqrt{2(1 - \cos t)}
= 2|\sin \tfrac{t}{2}|.
$$
If $0 \leq t \leq 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Manipulating a Summation Series
Summation: How does$$\begin{align*} & \frac 12\sum\limits_{k=1}^n\frac 1{2k-1}+\frac 12\sum\limits_{k=1}^n\frac 1{2k+1}-\frac 12\sum\limits_{k=1}^n\frac 1k+\frac n{2n+1}\tag1\\ & =\sum\limits_{k=1}^n\frac 1{2k-1}-\frac 12\sum\limits_{k=1}^n\frac 1k\tag2\\ & =\sum\limits_{k=1}^{2n}\frac ... | *
*Note that
$$\sum_{k=1}^n\frac{1}{2k+1}=\sum_{k=2}^{n+1}\frac{1}{2k-1}=\sum_{k=1}^{n}\frac{1}{2k-1}-1+\frac{1}{2n+1}$$
Therefore,
\begin{align*}
\frac{1}{2}\sum_{k=1}^n\frac{1}{2k-1}+\frac{1}{2}\sum_{k=1}^n\frac{1}{2k+1}+\frac{n}{2n+1}&=\sum_{k=1}^n\frac{1}{2k-1}-\frac{1}{2}+\frac{1}{2(2n+1)}+\frac{n}{2n+1}\\
&=\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Calculate $\sum_{n=1}^{\infty}{c_n\over n}$ when $c_1=2, c_2=1, c_3=-3, c_{n+3}=c_n, n\in \mathbb{N}$ Let $(c_n)$ be defined as $c_1=2, c_2=1, c_3=-3, c_{n+3}=c_n, n\in \mathbb{N}$. I want to get $\sum_{n=1}^{\infty}{c_n\over n}$. I would appreciate any suggestions how to start on this. It is my suspicion that I would ... | Break the sum into three pieces
\begin{eqnarray*}
S=\sum_{n=0} ^{\infty} \left( \frac{2}{3n+1} + \frac{1}{3n+2} +\frac{-3}{3n+3} \right) \\
\int_0^1 x^n dx = \frac{1}{n+1}
\end{eqnarray*}
Now use the above formula ... interchange the sum & integral & one has ...
\begin{eqnarray*}
S= \int_0^{1} \frac{2+x-3x^2}{1-x^3} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Fraction issue, I don't know why I'm having this result. I don't know how to get this result: $\frac{38}{17}$
This is the equation, can anyone explain why?
$(\frac{x}{4}) - (x - \frac{5}{6}) = (1 + \frac{2(x-5)}{3})$
I did this : LCM $= 12$.
Then: $$3(x) - 2(x-5) = 4(1 + 2(x-5)$$
$$3x - 2x + 10 = 4(1 + 2x - 10)$$
$$3... | $$ \frac{x}{4} - \left(x - \frac{5}{6}\right) = 1 + \frac{2(x-5)}{3} $$
First add the $x/4$ and $-x$ on the LHS and distribute the $2/3$ on the RHS,
$$ \frac{-3}{4}x + \frac{5}{6} = 1 + \frac{2}{3}x - \frac{10}{3} $$
Move all the constants to the LHS and $x$'s to the RHS,
$$ \frac{5}{6} - \frac{6}{6} + \frac{20}{6} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2308286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Volume of Rotation Let $W_n$ be the region in an $xy$-plane that’s bounded by the $y$-axis and the parabola $x=y^2-n^2$.
Find the value of $n$ for which the volume obtained by rotating $W_n$ about the $y$-axis is precisely 16 times the volume obtained by rotating $W_n$ about the $x$-axis.
| Lets start with the y-axis revolution.
We are integrating from $-n$ to $n$.
We get $2π\displaystyle \int_0^{n}(n^2-y^2)^2\,dy = \frac{16}{15}πn^5$
I put a $2$ in front for symmetry and ease in calculation.
Next, we do x-axis revolution. This one is a little bit tricky, we have to rewrite in terms of $x$.
$x=y^2-n^2$ tu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2312343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determine the minimal polynomial of $\alpha$ over $\Bbb Q(\sqrt{-2})$ Let $\alpha$ be a root of the polynomial $x^4-4x^2+2.$
How should I determine the minimal polynomial of $\alpha$ over $\Bbb Q(\sqrt{-2})$?
| Let $\alpha$ be a root of the polynomial $x^4-4x^2+2$
$$x^4-4x^2+2$$
$$\left(x^2 - \sqrt{2} -2\right) \left( x^2 + \sqrt{2} -2\right)$$
$$\left(\sqrt{2 - \sqrt2} - x\right) \left(\sqrt{2 + \sqrt2} - x\right) \left(\sqrt{2 - \sqrt2} + x\right) \left(\sqrt{2 + \sqrt2} + x\right)$$
Then
$$\alpha = \left(\sqrt{2 \pm \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2313315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Understanding proof of period of sin(ax/b) + cos(cx/d) I've been trying to understand this proof for determining that the period of $\sin(\frac{ax}{b}) + \cos(\frac{cx}{d})$ where $a, b, c, d$, are integers and $a \over b$, $c \over d$ are in lowest terms is $\frac{2 \pi *lcm(b,d)}{gcf(a, c)}$. The proof goes like this... | $\DeclareMathOperator{\lcm}{lcm}$It may help to work instead with the $1$-periodic functions
$$
C(x) = \cos(2\pi x),\qquad
S(x) = \sin(2\pi x),
$$
so that $S(\frac{a}{b}x) + C(\frac{c}{d}x)$ is $\ell$-periodic if and only if $\sin(\frac{a}{b}x) + \cos(\frac{c}{d}x)$ is $2\pi\ell$-periodic.
The question is, what is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2314163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the orthogonal projection of $(2,1,3)$. $V=\mathbb R^3,u=(2,1,3) $ and $W$={$(x,y,z):x+3y-2z=0$}.
Vectors ortogonal to $W$=Span{$(1,3,-2)$},Vectors on $W$=Span{$(-1,1,1)$}.
So,orthogonal basis is {$(1,3,-2),(-1,1,1)$}.Then,orthonormal basis is
{$\frac{1}{\sqrt {14}}(1,3,-2),\frac{1}{\sqrt {3}}(-1,1,1)$}.
So or... | If $W = \{(x,y,z): x+3y-2z=0\}$, then $x=2z-3y$, and every element of $W$ looks like
$$\left(\begin{array}{c} x \\ y \\ z \end{array}\right)
\ \ = \ \ \left(\begin{array}{c} 2z-3y \\ y \\ z \end{array}\right)
\ \ = \ \ y\left(\begin{array}{c} -3 \\ 1 \\ 0 \end{array}\right)
+z\left(\begin{array}{c} 2 \\ 0 \\ 1 \end{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Maximization of multivariable function $ M = \frac{a+1}{a^{2}+ 2a+2} + \frac{b+1}{b^{2}+ 2b+2} + \frac{c+1}{c^{2}+ 2c+2}$ subject to constraint This is purely out of curiosity. I have this set of a high school entrance Math exam in Vietnam (a special high school for gifted kids, the exam was held 3 days ago, maybe 4, t... | If $a=b=c=\sqrt3-1$ then we get a value $\frac{3\sqrt3}{4}$.
We'll prove that it's a maximal value.
Indeed, let $x=\frac{a+1}{\sqrt3}$, $y=\frac{b+1}{\sqrt3}$ and $z=\frac{c+1}{\sqrt3}$.
Hence, the condition gives $x+y+z=3xyz$ and we need to prove that
$$\sum_{cyc}\frac{x}{3x^2+1}\leq\frac{3}{4}$$ or
$$\sum_{cyc}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Conversion of $x^2-2xy-y^2-x+y=0$ to $Ax^2-BY^2=C$ to solve diophantine equation. Given $x^2-2xy-y^2-x+y$ we see that $\Delta=(-2)^2-4(1)(-1)=4+4=8>0$, therefore it is a hyperbola, I was required to convert it to $AX^2-BY^2=C$, so first I removed the $xy$ term by noticing that rotation of $-\pi/8$ converted it to
$$\sq... | Discriminant of the quadratic (in terms of $x$ for example) must be a perfect square, thus $$(2y+1)^2-4(-y^2+y)=z^2$$or $$8y^2+1=z^2$$Its a Pell equation now.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Infinite zeros in infinite series The problem:
Given that
$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \ldots $$
Prove
$$\frac{\pi}{3} = 1 + \frac{1}{5} - \frac{1}{7} - \frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \ldots$$
My solution:
We know
$$
\begin{align}
\frac{\pi}{4} & = 1 - \f... | It is a rewarding exercise to write, or read, detailed rigorous proofs of some simple, "obvious" results.
The definition of $x=\sum_{j=1}^{\infty}y_j$ is that $x=\lim_{n\to \infty}S_n$ where $S_n=\sum_{j=1}^ny_j.$ A useful way to state (or define) that the sequence $S=(S_n)_n$ converges to $x$ is that for any $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 1
} |
If $p \equiv 1 \pmod{4}$, then $p \nmid a^2+3b^2$? Suppose that $a, b$ are co-prime numbers and $p$ is a prime number in the form $4k+1$. Can $3a^2+b^2$ have a prime divisor like $p$?
I really don't know how to think about this! Please help!
| Suppose we have $$a^2\equiv -3b^2\pmod p$$
Now, since $\gcd(a,b)=1$ we can exclude the case where one, hence both, of $a,b$ are divisible by $p$. Thus we can divide by $b^2$ to see that $-3$ must be a square $\pmod p$.
Assume that $p\equiv 1 \pmod 4$. Quadratic reciprocity then tells us that $-3$ is a square $\pmod p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
need help for proof of $\sum_{i=1}^{n} \left(3 \cdot (\frac{1}{2})^{i-1} +2 \, i \right)$ So I'm asked to start with:
$$\sum_{i=1}^{n} \left(3 \cdot (\frac{1}{2})^{i-1} +2 \, i \right)$$
My notes say to start by taking out the 3, use i-1 and make it multiply by $\frac{1}{2}$, and then transforming the last part by tak... | hint
$$S (x)=\sum_{i=1}^n3x^{i-1}=3\frac {x^n-1}{x-1} $$
$$\sum_{i=1}^n 2i=2 (1+2+3+... n)=2\frac {n (n+1)}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Integrate the expression $\int\frac{x}{1+\sin x}\ \text dx$
Integrate the expression $\int\frac{x}{1+\sin x}\ \text dx$
I really do not see any identity working here. I tried rewriting in terms of $\cos x$ and got $$\int\frac{\frac{x}{\cos x}}{\frac{1}{\cos x}+\frac{\sin x}{\cos x}}\ \text dx$$
| $$I = \int\frac{x}{1+\sin x}\, dx$$
Multiply by the conjugate of $1 + \sin x$
$$\int\frac{x}{1+\sin x}\cdot \frac{1-\sin x}{1-\sin x}\, dx = $$
$$\int\frac{x}{1+\sin x}\cdot \frac{1-\sin x}{1-\sin x}\, dx = \int\frac{x-x\sin x}{1-\sin^2 x}\, dx$$
$$\int\frac{x-x\sin x}{\cos^2 x}\, dx$$
$$\int\frac{x}{\cos^2 x}\, dx- \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Limits problem $ \lim_{x\to \pi^+/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$ $$ \lim_{x\to \pi^+/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$$
I tried to solve this by L'hostipal's rule..but that doesn't give a solution..appreciate if you can give a clue..
$ \lim_{x\to \pi^+/2} \frac{\sqrt{1+\cos(2x)}}{\... | \begin{align}
\lim_{x\to \frac{\pi^+}{2}} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}&= \lim_{x\to \frac{\pi^+}{2}} \frac{\sqrt{2\cos^2x}}{\sqrt{\pi}-\sqrt{2x}}\\
&= \lim_{x\to \frac{\pi^+}{2}} \frac{-\sqrt{2}\cos x}{\sqrt{\pi}-\sqrt{2x}}\\
&= \lim_{x\to \frac{\pi^+}{2}} \frac{-\sqrt{2}\cos x(\sqrt{\pi}+\sqrt{2x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find extremas of $f(x,y) = xy \ln(x^2+y^2), x>0, y>0$ As the title says I need to find extreme values(maximum and minimum) of
$$f(x,y) = xy \ln(x^2+y^2), x>0, y>0$$
I don't understand how to find critical points of this problem.
I start with finding partial derivative and set derivatives equal to zero. And that is wher... | It's easier to work with polar coordinates $x = r \cos \theta, y = r \sin \theta$ where (in your case) $r > 0$ and $0 < \theta < \frac{\pi}{2}$. Then we have
$$ f(x,y) = xy \ln(x^2 + y^2) = r \cos \theta r \sin \theta \ln(r^2) = r^2 \ln r \sin(2 \theta) = g(r,\theta). $$
The partial derivatives of $g$ are given by
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Solving equation of type $a\cos x+b\cos y-c=0$ and $a\sin x+b\sin y-d=0$ Here's the questions
There are two equations:
$a\cos x+b\cos y-c=0 $ and $ a\sin x+b\sin y-d=0$ .
For instance
What is the value of $x$ and $y$ in following question?
$$2\cos x+3\cos y-2=0$$
$$2\sin x+3\sin y-8=0$$
| just a hint
From the first equation
$$2\cos (x)=2-3\cos (y) $$
and from the second
$$2\sin (x)=2-3\sin (y) $$
thus
$$(2\sin (x))^2+(2\cos (x))^2=$$
$$4=8-12\cos(y )-12\sin(y)+9$$
$$\cos(y)+\sin (y)=\frac {13}{12}$$
$$\sqrt {2}\cos (y-\frac {\pi}{4})=\frac {13}{12}$$
You can finish.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Prove $\cos6x=32\cos^{6}x-48\cos^{4}x+18\cos^{2}x-1 $ So far I've done this:
LHS $ =\cos^{2}3x-\sin^{2}3x$
$={(4\cos^{3}x-3\cos{x})}^2 -{(3\sin{x}-4\sin^{3}x)}^2$
$=16\cos^{6}x+9\cos^{2}x-24\cos^{4}x-9\sin^{2}x-16\sin^{6}x+24\sin^{4}x$
I can tell I'm going in the right direction but how should I proceed further?
EDIT ... | Note that you can use the identity $\cos^2 x+\sin^2 x=1$(or alternatively, $\sin^2 x=1-\cos^2 x$).
Take each side of the equation to the power of $2,3$ to get $$9 \sin^2 x=9-9\cos^2 x$$
$$24\sin^4 x=24-48 \cos x^2+24 \cos^4 x$$
$$16\sin^6 x=16-48 \cos^2 x+48 \cos^4 x-16\cos^6 x$$
Tedious, but it probably will work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
If $b>a,$ find the minimum value of $|(x-a)^3|+|(x-b)^3|,x\in R$ If $b>a,$ find the minimum value of $|(x-a)^3|+|(x-b)^3|,x\in R$
Let $f(x)=|(x-a)^3|+|(x-b)^3|$
When $x>b,f(x)=(x-a)^3+(x-b)^3$
When $a<x<b,f(x)=(x-a)^3-(x-b)^3$
When $x<a,f(x)=-(x-a)^3-(x-b)^3$
I am stuck here.The answer given in my book is $\frac{(b-a)... | Let $a<x<b$.
Hence, since $f(x)=x^3$ is a convex function, by Jensen we obtain:
$$|(x-a)^3|+|(x-b)^3|=(x-a)^3+(b-x)^3\geq2\left(\frac{x-a+b-x}{2}\right)^3=\frac{(b-a)^3}{4}$$
Let $x\geq b$. Hence,
$$|(x-a)^3|+|(x-b)^3=(x-a)^3+(x-b)^3\geq(b-a)^3>\frac{(b-a)^3}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.