Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods???
i only know quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ i tried many algebraic manipulations and i get $(x^2+1)^2=3(x^3+x)$, so can we have solution ... | we have $x^4-3x^3+2x^2-3x+1=0$ and we see that $x=0$ is not a solution, since we can write
$x^2+1/x^2-3(x+1/x)+2=0$
setting $t=x+1/x$ we get $x^2+1/x^2=t^2-2$ and we have a quadratic equation
$t^2-3t=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1023338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Evaluating $\int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2 x}\right)dx$ Evaluate
$$\displaystyle \int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2 x}\right)dx$$
$\bf{My\; Try::}$
Let $$\begin{align}I &= \int e^{x\sin x+\cos x}\left(\frac{x^4\cos^3 x-x\sin x+\cos x}{x... | Of course $\left( e^{x\sin x + \cos x}\right)' = x\cos x . e^{x\sin x + \cos x}$. But in the integrand we have terms such as $x^2\cos x.e^{...}$. So let's try
$$\left(x e^{x\sin x + \cos x}\right)' = (x^2 \cos x + 1)e^{...}$$
Hence the integral
$$I = x e^{x\sin x + \cos x} + \int e^{x\sin x + \cos x} \left( \frac{-\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1024099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Optimization problem - Trapezoid under a parabola recently I've been working on a problem from a textbook about Optimization. The result that I get is $k = 8$, even thought the answer from the textbook is $k = \frac{32}{3}$
The problem follows:
--
The x axis interepts the parabola $12-3x^2$ at the points $A$ and $B$, a... | An alternative method:
Once you get to expressing $A$ in terms of $k$, substitute $k=12\sin^2\theta$ to help avoid messy surds.
$$\begin{align}\\
A&=k(2+\sqrt{4-\frac k3})\\
&=12\sin^2\theta(2+\sqrt{4-4\sin^2\theta})\\
&=24\sin^2\theta(1+\cos\theta) \\
\frac{dA}{d\theta}&=24[2\sin\theta\cos\theta(1+\cos\theta)+\sin^2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1025897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find the minimum value of the expression $P=\frac{x^2}{\sqrt{x^3+8}}+\frac{y^2}{\sqrt{y^3+8}}+\frac{z^2}{\sqrt{z^3+8}}$ Let $x,y,z$ be positive real number such that $xy+yz+zx=3$. Find the minimum value of the expression $$P=\frac{x^2}{\sqrt{x^3+8}}+\frac{y^2}{\sqrt{y^3+8}}+\frac{z^2}{\sqrt{z^3+8}}$$
| It is easy to guess from the symmetry that $x=y=z=1$ should give the minimum of $1$, and we will try proving this. By Cauchy-Schwarz Inequality:
$$\left(\sum_{cyc} \frac{x^2}{\sqrt{x^3+8}} \right)\left(\sum_{cyc} \sqrt{x^3+8} \right) \ge (x+y+z)^2 $$
We will use $\sum $ to denote cyclic sums. From the above, if we sho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
If the number $x$ is algebraic, then $x^2$ is also algebraic Prove that if the number $x$ is algebraic, then $x^2$ is also algebraic. I understand that an algebraic number can be written as a polynomial that is equal to $0$. However, I'm baffled when showing how $x^2$ is also algebraic.
| We have $P(x)=0$, where $P$ is some rational polynomial (that is, the coefficients are rational numbers). Break $P$ up into the terms with odd exponents and the terms with even exponents. For example, if $P(x)=x^4+x^3+5x^2+x+4$, then we would break it up as $(4+5x^2+x^4)+(x+x^3)$. The term with even exponents can be vi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1026778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
Approximate to whole number without calculator Approximate this number to the nearest whole number without a calculator:
$2^{\sqrt{5}}$
I don't know how to do this problem. Can you help me? My answer key says $5$.
Also, how do I approximate $2^\pi$ and $3^e$? The answers are $9$ and $20$, respectively.
| i can get $4 < 2^{\sqrt 5} < 5.$ use the fact that $\sqrt x$ is concave down so the tangent approximation $\sqrt 5 = \sqrt {4 + 1)} < \sqrt 4 + {1 \over 2 \sqrt 4} = {5 \over 4}.$ so we have $2 < \sqrt 5 < {5 \over 4}$ and $2 = {512 \over 256} < {625 \over 256}$ gives us $2^{1/4} < {5 \over 4}.$
so far we have, $2 < ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1028661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why is Implicit Differentiation needed for Derivative of $y = \arcsin (2x+1)$? my function is: $y = \arcsin (2x+1)$ and I want to find its derivative.
My approach was to apply the chain rule:
$$y' = \frac{dg}{du} \frac{du}{dx}$$
with $g = \arcsin(u)$ and $u = 2x+1$.
$$g' = \frac{1}{\sqrt{1-u^2}}.$$
${u}' = 2$.
My so... | Your solution is correct; they just simplified it further:
\begin{align*}
\frac{2}{\sqrt{1 - (2x + 1)^2}}
&= \frac{2}{\sqrt{1 - (4x^2 + 4x + 1)}} \\
&= \frac{2}{\sqrt{-4x^2 - 4x}} \\
&= \frac{2}{\sqrt{4(-x^2 - x)}} \\
&= \frac{2}{\sqrt{4}\sqrt{-x^2 - x}} \\
&= \frac{2}{2\sqrt{-x^2 - x}} \\
&= \frac{1}{\sqrt{-x^2 - x}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1029724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$
Evaluate
$$
\int_{0}^{1} \arctan^{2}\left(\, x\,\right)
\ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x
$$
I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got
$$
-\int^{\pi/4}_{0}\theta^{2}\,{\ln\left(\, 2\sin^{2}\left(\,\theta\,... | By the way, there is a closed-form antiderivative (that could be proved by differentiation):
$$\int\arctan^2x\cdot
\ln\left(\frac{1+x^2}{2x^2}\right)\,dx=\\
\frac16\left[3 i \left\{\left(2 \operatorname{Li}_2(i
x)-2 \operatorname{Li}_2(-i x)+\operatorname{Li}_2\left(\frac{2
x}{x+i}\right)-\operatorname{Li}_2\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1032483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 5,
"answer_id": 2
} |
Using mathematical induction to show that for any $n\ge 2$ then $\prod_{i=2}^n\bigl(1-\frac{1}{i^2}\bigr)=\frac{n+1}{2 n}$ I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ... | Base case p(2):
LHS: $$\prod_{i=2}^2 \left(1-\frac{1}{i^2}\right) = 1-\frac{1}{2^2} = \frac{3}{4}$$
RHS: $$\frac{2+1}{2\cdot2} = \frac{3}{4}$$
Now assume p(k):
$$\prod_{i=2}^k\left(1-\frac{1}{i^2}\right)=\frac{k+1}{2k}$$
$$\Rightarrow \prod_{i=2}^{k+1}\left(1-\frac{1}{i^2}\right) = \frac{k+1}{2k} \left(1-\frac{1}{(k+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1033558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$ I have a function:
$$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$
With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can have an invers? Or is there another/simplier way to do i... | This is probably conceptually the simplest method.
For $\frac{a}{b}=\frac{c}{d}$, we always have $\frac{a+b}{a-b}=\frac{c+d}{c-d}$, provided that $a-b\not=0$ and $c-d\not=0$.
$$y = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$
$$\frac{y}{1} = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$
$$\frac{y+1}{y-1} = \frac{1-\sqrt{x}+(1+\sqrt{x})}{1-\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1034887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$ If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation
$$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$
lying inside the unit circle $\vert z\vert$=1, satisfies which inequality?
A)$\vert z\vert$ < $\frac{2}{3}$
B)$\vert z... | I am assuming that by $\sin \theta_1 z^3$ our OP Jackie means $z^3 \sin \theta_1$ and so forth; with this understanding, we have the given equation
$z^3 \sin \theta_1 + z^2 \sin \theta_2 + z \sin \theta_3 + \sin \theta_4 = 3; \tag{1}$
taking absolute values and using the triangle inequality (several times) yields
$3 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1035351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $\lim_{x\rightarrow \infty}\sqrt{x^2+x+1} -\lfloor \sqrt{x^2+x+1 }\rfloor\;,$ Evaluation of $\displaystyle \lim_{x\rightarrow \infty}\sqrt{x^2+x+1} -\lfloor \sqrt{x^2+x+1 }\rfloor\;,$ where $\lfloor x \rfloor$ represent floor function of $x$.
$\bf{My\; Try}::$
$\bullet\; $If $x\in \mathbb{Z}\;,$ and $x\r... | I think this problem is much more easily solved in slightly greater generality. If $f$ is a continuous function such that $\lim_{x \rightarrow \infty} f(x) = \infty$ then as $x \rightarrow \infty$, for arbitrarily large $x$, $f(x)$ is an integer and for any $\varepsilon < 1$, for arbitrarily large $x$, $f(x)$ is an int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1036936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Find the value of this infinite product $\prod_{n=1}^{\infty }\left [ 1+\frac{1}{a_{n}} \right ]$? Where $a_{1}=1$ and $a_{n+1}= \left ( n+1 \right )\left ( 1+a_{n} \right )$,
$\forall \left ( n\epsilon \mathbb{N} \right )$
I have found that for this infinite product to be convergent, the series $\sum \ln \left ( 1+\f... | Note that
$$a_{n+1}=(n+1)(1+a_n)\Rightarrow \frac{a_{n+1}}{1+a_{n}}=n+1$$
Consider the finite product:
$$\mathcal{P}_n=\prod_{n=1}^{N}(1+\frac{1}{a_n})=\prod_{n=1}^{N}\frac{1+a_n}{a_n}=\prod_{n=1}^{N}\frac{a_{n+1}}{(n+1)a_n}=\prod_{n=1}^{N}\frac{1}{(n+1)}\cdot \prod_{n=1}^{N}\frac{a_{n+1}}{a_n}=\frac{1}{N!}\cdot\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1038616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$ How can you determine which one of these numbers is bigger (without calculating):
$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
| What's wrong with just logging? Logarithm is a monotone increasing function, so the inequality sign stays the same.
First log, the multiply both sides by 2 and 3. LHS becomes $\log (\frac{1}{2})^2$, right $\log(\frac{1}{3})^3$. Now exponentiate. LHS is $\frac{1}{2} \cdot \frac{1}{2} \cdot 1$, RHS is $\frac{1}{3} \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1041684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 9,
"answer_id": 8
} |
$4$ dice sum probability What is the probability that, when $4$ dice are rolled, we can choose two of them such that the sum of the numbers on the upper faces is $7$?
I get $139 \over 216$ but I'm not sure I'm correct.
| It is convenient to assume that the dice were rolled one at a time, and that we recorded the results as they came in. So there are $6^4$ equally likely outcomes. We count the favourables.
First we count the number of strings that have $6$ and a $1$. It is easier to count the strings that don't. There are $5^4$ string... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is it correct? $\tan x + \cot x \ge 2$ proof The question is to prove $\tan x + \cot x \ge 2$ when $x$ is an acute angel.
This is what I did
$$\begin{align}
\tan x + \cot x &\ge 2\\
\frac{1}{\sin x \cos x} &\ge 2\\
\left(\frac{1}{\sin x \cos x}\right) - 2 &\ge 0\\
\left(\frac{1 - 2\sin x \cos x}{\sin x \cos x}\right) &... | It is quite easy...
$\tan x + \cot x$
= $\tan x + \cot x - 2 \sqrt {\tan x \cot x} + 2 \sqrt{\tan x \cot x}$
=$(\sqrt {\tan x} -\sqrt {\cot x})^2+ 2$ [because $\tan x \cot x =1$]
Since square of any number is greater than or equal to zero, we can write,
$(\sqrt {\tan x} -\sqrt {\cot x})^2 \ge 0$
So, $(\sqrt {\tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 4
} |
Evaluating $\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$ $$\int\frac{4x^3-3x^2+6x-27}{x^4+9x^2}dx$$
this integral get very messy. Can I get a step by step breakdown of solving?
|
No Step by step solution, but just a hint
By reducing to partial fractions we get
$$\frac{4x^3-3x^2+6x-27}{x^4+9x^2}=\frac{Ax+ B}{x^2+9}+\frac{C}{x^2}+\frac Dx$$
$$Ax^3+Bx^2+Cx^2+9C+Dx^3+9Dx=4x^3-3x^2+6x-27$$
$$9C=-27\implies C=-3$$
$$9D=6\implies D=\frac23$$
$$A+D=4\implies A=-\frac{10}{3}$$
$$B+C=-3\implies B=0$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $x$, $y$, $z$ are in arithmetic progression, show that $\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $ Show that if $x, y,$ and $z$ are consecutive terms of an arithmetic sequence, and $\tan y$ is defined, then
$$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y.
$$
I'm not... | Lets define the common ratio $r$
Using the 4 Trigonometric Conversion functions, we get:
\begin{eqnarray*} \sin(y+r) &=& \sin y \cos r + \cos y \sin r \\ \sin(y-r) &=& \sin y \cos r - \cos y \sin r \\ \cos(y+r) &=& \cos y \cos r - \sin y \sin r \\ \cos(y-r) &=& \cos y \cos r + \sin y \sin r
\end{eqnarray*}Since $x = y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How come $\frac{1}{\cos x} = 1 + \frac{x^{2}}{2} + o(x^{2})$ as $x \to 0$? Since
$$\cos x = 1 - \frac{x^{2}}{2} + o(x^{3})$$
as
$x \to 0,$
we have
$$\frac{1}{\cos x} = \frac{1}{1-\frac{x^{2}}{2} + o(x^{3})} = 1 + \frac{x^{2}}{2} + o(x^{3}) + o(\frac{-x^{2}}{2} + o(x^{3})).$$
But I do not see how to write the term on ... | You may recall that
$$
\frac{1}{1-u}=1+u+O(u^2), \quad u \to 0.
$$
Thus
$$
\frac{1}{\cos x} = \frac{1}{1-\frac{x^{2}}{2} + o(x^{3})} = 1 + \frac{x^{2}}{2} + o(x^{3}) + O(\frac{-x^{2}}{2} + o(x^{3}))^2
$$ then use
$$
O(\frac{-x^{2}}{2} + o(x^{3}))^2 =o(x^{3}).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can I get a little integral help? Find the integral $$\int_0^3 \sqrt {y+1}\, dy$$
| $$\int_0^{3} \sqrt{y+1} dy = \int_0^{3} (y+1)^{\frac{1}{2}}d (y + 1) = \frac{2}{3} \int_0^{3} \frac{3}{2}(y+1)^{\frac{1}{2}}d (y + 1) = \left. \frac{2}{3}(y+1)^{\frac{3}{2}} \right|_0^3 = \frac{2}{3} 4 ^{\frac{3}{2}} - \frac{2}{3}1^{\frac{3}{2}} = \frac{16}{3} - \frac{2}{3} = \frac{14}{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Show that $\sin45°+\sin15°=\sin75°$ Steps I took:
1) Finding the value of the left hand side
$$\sin45=\sin\frac { 90 }{ 2 } =\sqrt { \frac { 1-\cos90 }{ 2 } } =\sqrt { \frac { 1 }{ 2 } } =\frac { \sqrt { 2 } }{ 2 } $$
$$\sin15=\sin\frac { 30 }{ 2 } =\sqrt { \frac { 1-\cos30 }{ 2 } } =\sqrt { \frac { 1-\frac { \sqrt... | We have:
$$\frac { \sqrt { 2 } +\sqrt { 2-\sqrt { 3 } } }{ 2 }$$
By squaring the numerator and denominator:
$$\frac {\left(\sqrt{2}+\sqrt{2-\sqrt{3}}\right)^2}{4} = \frac{2+2-\sqrt{3}+ 2\sqrt{4-2\sqrt3}}{4} = \frac{4-\sqrt{3}+2\sqrt{4-2\sqrt3}}{4}$$
Next we must find:
$$\sqrt{4-2\sqrt3}$$
Let's call the answer $a-\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Binomial Expansion with fractional or negative indices Question:
Expand the function $\frac{2}{(2x - 3)(2x+1)}$ in a series of powers of $x$ up to $x^2$. State the set of values of $x$ for which this expansion is valid.
I've come across this question and would like to ask how most of you would tackle it. I've never se... | The Binomial Theorem for negative powers says that for $|x| < 1$
$$(1+x)^{-1} = 1 - x + x^2 + \mathcal{o}(x^2)$$
Therefore we have:
$$\frac 2{(2x-3)(2x+1)} = \frac 1{2(2x-3)} - \frac 1{2(2x+1)} = -\frac 16\left(1-\frac 23x\right)^{-1} - \frac 12\left(1+2x \right )^{-1} = -\frac16\left(1 + \frac 23x + \frac 49x^2\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving $\sin\theta -1=\cos\theta $
Solve$$\sin\theta -1=\cos\theta $$
Steps I took to solve this:
$$\sin^{ 2 }\theta -2\sin\theta +1=1-\sin^2\theta $$
$$2\sin^{ 2 }\theta -2\sin\theta =0$$
$$(2\sin\theta )(\sin\theta -1)=0$$
$$2\sin\theta =0, \sin\theta -1=0$$
$$\quad \sin\theta =0, \sin\theta =1$$
$$\theta =0+\pi ... | When you square both sides of an equation, you may introduce extraneous solutions. Therefore, you must check that your solutions satisfy the original equations (a good idea in any case).
When you squared the equation $\sin\theta - 1 = \cos\theta$, you discovered that
the resulting equation was satisfied when $\sin\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Show that this two variable limit don't exist Prove that the following limit doesn't exist: $$\lim_{(x,y)\to(1,0)}\frac{\sin(1-x^2+y^2)-y}{xy}$$
I guess I need to find different values for different directions.
| First direction: $y=1-x,$ second direction: $y=-(1-x).$
If $y=1-x$ then
$$\begin{gather}\lim_{\substack{{(x,y)\to(1,0)}\\{y=1-x}}}\frac{\sin(1-x^2+y^2)-y}{xy}=
\lim_{x\to{1}}\frac{\sin((1-x)(1+x)+(1-x)^2)-(1-x)}{x(1-x)}=\\
=\lim_{x\to{1}}\frac{\sin{\left((1-x)(1+x+1-x)\right)-(1-x)}}{x(1-x)}=
\lim_{x\to{1}}\frac{\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Confusion about a Linear Transformation question. Let $\beta := [M_1, M_2, M_3, M_4]$ be the ordered basis of $R^{2×2}$ defined by:
$$ M_1 := \begin{pmatrix}
1 & 0\\
0 & 0 \end{pmatrix},
M_2 := \begin{pmatrix}
0 & 1\\
0 & 0 \end{pmatrix},
M_3 := \begin{pmatrix}
0 & 0\\
1 & 0 \end{pmatrix},
M_4 := \begin{pmatrix}
0 & 0... | Your transformation is from $\mathbb{R}^{2\times 2}$ to $\mathbb{R}^{2\times 2}$; and from the given question, it looks like both the domain and the range spaces are represented using the same basis that is $\left\{b_1=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},b_2=\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix},b_3=\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determine when the system has a) no solution, b) 1 solution and c) infinitely many solutions This question is not for an assignment, it was on the midterm and I am interested in figuring out how to solve it before the final exam.
cheers,
Determine when the system has a) no solution, b) 1 solution and c) infinitely man... | You have arrived to this (which is followed by some not very reader friendly part, which I am ignoring EDIT: In the meantime, N. F. Taussig fixed it):
$$\left(\begin{array}{ccc|c}
1&1&7&-7\\
0&-1&-3&2\\
0&-1&7-(a^2+1)&-7-3a
\end{array}\right)=
\left(\begin{array}{ccc|c}
1&1&7&-7\\
0&-1&-3&2\\
0&-1&6-a^2&-7-3a
\end{arra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1062990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the Viète formula I know that the Viète formula for the equation $ax^2+bx+c=0$ is:
$$x_1+x_2=-\frac{b}{a}$$
$$x_1x_2=\frac{c}{a}$$
But I didnt know which are the formula for the equation $ax^3+bx^2+cx+d=0$.
Please help me. Thanks.
| Soppose that $x_1, x_2, x_3$ is zeros of equation $ax^3+bx^2+cx+d=0$. Now we have
$$ax^3+bx^2+cx+d=a(x-x_1)(x-x_2)(x-x_3)$$
$$=a(x^2-xx_1-xx_2+x_1x_2)(x-x_3)$$
$$=a(x^3-x^2x_1-x^2x_2+xx_1x_2-x^2x_3+xx_1x_3+xx_2x_3-x_1x_2x_3)$$
$$=a(x^3-x^2(x_1+x_2+x_3)+x(x_1x_2+x_1x_3+x_2x_3)-x_1x_2x_3)$$
For the F. Viéte formula we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1063683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
deriving the sum of $x^n/(n+2)^2$ I am writing a research paper and I have stumbled upon an issue.
I have to evaluate
$$\sum_{n=1}^{\infty} \frac{x^n}{(n+2)^2}$$
Here is what I did:
$$ \sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$
$$\sum_{n=1}^{\infty} x^{n+1} = \frac{x^2}{1-x}$$
Integrate once with respect to $x$.... | Somewhere in your question, you write "Integrate once with respect to $x.$". In the equation following that phrase, you have a $\frac{3}{2}$ on the r.h.s. That's superfluous. The derivatives of the l.h.s. and the r.h.s are equal, and setting $x=0,$ we find that the constant of integration is $0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
prove that $\sqrt{2} \sin10^\circ+ \sqrt{3} \cos35^\circ= \sin55^\circ+ 2\cos65^\circ$ Question:
Prove that: $\sqrt{2} \sin10^\circ + \sqrt{3} \cos35^\circ = \sin55^\circ + 2\cos65^\circ$
My Efforts:
$$2[\frac{1}{\sqrt{2}}\sin10] + 2[\frac{\sqrt{3}}{2}\cos35]$$
$$= 2[\cos45 \sin10] + 2[\sin60 \cos35]$$
| Hints:
$$\begin{align}&\bullet\;\;\sin(x+y)=\sin x\cos y+\sin y\cos x\\{}\\&\bullet\;\;\sin 45^\circ=\frac1{\sqrt2}=\frac{\sqrt2}2=\cos45^\circ\\{}\\&\bullet\;\;\cos x=\cos(-x)\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1067644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Proving that for all complex $z$, $\lim_{x\to0}\frac{1-\cos^{z}x}{x^2}=\frac{z}{2}.$ What do I need to study beforehand in order to prove it (not necessarily in only one way)? I found this sperimentally, at the moment we're beginning derivatives at school. By induction, I succeeded in proving $$\lim_{x\to0}\frac{1-\cos... | $$\frac{1-\cos^z x}{x^2}=\frac{1-\exp(z\log \cos x)}{x^2}=\frac{1-\exp(z\log \cos x)}{\log\cos x}\cdot\frac{\log\cos x}{x^2}$$
But since $\log\cos x\to 0$ when $x\to 0$,
$$\lim_{x\to 0}\frac{1-\cos^z x}{x^2}=-z\cdot\lim_{x\to 0}\frac{\log\cos x}{x^2}\stackrel{H}{=}z\cdot\lim_{x\to 0}\frac{\tan x}{2x}=\frac{z}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1068044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that $c_n = \frac1n \bigl(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \bigr)$ converges I want to show that $c_n$ converges to a value $L$ where:
$$c_n = \frac{\large \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}}{n}$$
First, it's obvious that $c_n > 0$.
I was abl... | Hint : $c_n$ is Cesàro summation of sequence $\{\frac{1}{\sqrt{n}}\}_{n \in \Bbb N}$
Cesàro summation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1070575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 0
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Forcing an absolute value of x after a square root operation Given the following two equations:
$$
f(x) = x \\
g^2(x) = 2x
$$
I need to find the $(x,y)$ coordinates for when they meet.
So after performing the square root operation, we have:
$$
f(x) = x \\
g(x) = \pm\sqrt{2\lvert x\rvert}
$$
So when trying to find the c... | \begin{align}
f(x)&=g(x)\\
x&=\pm\sqrt{2x}\tag{1}\\
x^2&=2x\tag{2}\\
x^2-2x&=0\\
x(x-2)&=0\\
x&=0,\,2
\end{align}
Now check if these are indeed solutions to the original equation $(1)$. (They are.)
Back to your proposed solution set. Note that $x=-2$ is not a solution of $(1)$ since the left-hand side is $-2$, but the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1070751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to use complex analysis to find the integral $\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$? How can I use complex analysis to solve the following:
$$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
| Start with
$$
z=e^{i\theta} \\
\log z=i\theta \\
\theta=-i\log z \\
d\theta=-\frac izdz
$$
Now we substitute $z$ into our integral.
$$
\oint\frac{-i}{z(1+\sin^2(-i\log z))}dz \\
$$
Analyzing just the $\sin$ part for a moment:
$$
\begin{align}
\sin(-i\log z)&=-\frac i2\left(e^{i(-i\log z)}-e^{-i(-i\log z)}\right)\\
&=-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions Let
$$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$
How to show that it converges with no use of trigonometric functions?
(trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be computed directly)
| here is a way to show that $\int_0^1 \frac{dx}{\sqrt{1-x^2}} = {\pi \over 2}$ without the use of trigonometric functions. i will use the fact the area of unit circle is $\pi.$ that is $\int_0^1 \sqrt{1-x^2}dx = {\pi \over 4}$ twice and integration by parts.
\begin{eqnarray}
{\pi \over 4} & = & \int_0^1 \sqrt{1-x^2}\ d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1074532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
How evaluate the following hard integrals? Prove:
$$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$
And $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{2\sin^2x}{\cos2x}}}dx=\frac{\pi}{192}[{\pi^2}+12\ln^22]$$
These integrals ... | Denote the first integral by $I$ and the second by $J$. Then,
$$\begin{aligned}
J=&\int_0^{\pi/4} x\left(\frac{\pi}{2}-\arctan\sqrt{\frac{\cos 2x}{2\sin^2 x}}\right)\,dx \\
=&\frac{\pi^3}{64}-I \,\,\,\,\,\,\,(1)
\end{aligned}$$
$I$ can be simiplified to:
$$I=\int_0^{\pi/4} x\arccos(\sqrt{2}\sin x)\,dx=\left(\frac{x^2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Discriminant of the polynomial $f(x)=4x^3-ax-b$
Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$.
How to prove that the discriminant of $f(x)=4x^3-ax-b$ is $a^3-27b^2$.
Any help would be appreciated.
| You should read about the sylvester Sylvester matrix.
To be precise, $f(x) = 4x^3-ax−b$ gives $f'(x) = 12 x^2 - a$. So the discriminant is the determinant
$$ \Delta(a,b) = \left|\begin{array}{ccccc}
4 & 0 & -a & -b & 0 \\
0 & 4 & 0 & -a & -b\\
12 & 0 & -a & 0 & 0\\
0 & 12 & 0 & -a & 0\\
0 & 0 &12 & 0 & -a
\end{array}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
In $\triangle ABC$ , find the value of $\cos A+\cos B$ The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy
$\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value of $\cos A+\cos B$ is $\dfrac{12}{7}$.
It took me a long time to solv... | $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}$=$\dfrac{2}{10!}$($^{10}C_1$)+$\dfrac{2}{10!}$($^{10}C_3$)+$\dfrac{1}{10!}$($^{10}C_5$)
$\implies$$\dfrac{1}{10!}$(2$\cdot$$^{10}C_1$+2$\cdot$$^{10}C_3+^{10}C_5$)
$\implies$$\dfrac{1}{10!}$(2$\cdot$10+2$\cdot$120+252)=$\dfrac{512}{10!}$=$\dfrac{2^9}{10!}$
$\implies$$\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find all the prime numbers that satisfy the following conditions There was a brainteaser in the Science Magazine from University of Hong Kong which is as follow:
Find all the prime numbers $p$ such that $\sqrt{\frac{p+7}{9p-1}}$ is rational.
I tried a few numbers and it seems to suggest that $11$ is a suitable candidat... | Let the rational number be $a/b$ in lowest terms. Rearrange the expression to $$p=\frac{a^2+7b^2}{9a^2-b^2}$$
Let $q$ be a prime factor of both numerator and denominator. Either $3a+b$ or $3a-b$ is a multiple of $q$, so $b=\pm3a+nq$ for some whole number $n$. Then $a^2+7b^2=a^2+7(9a^2+mq)$ for another whole number $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1078489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Difference of the roots of quadratic formula I have a question to solve with roots quadratic formula that is ,
$$a^3 + b^3 = (a+b)(a^2-ab+b^2)$$
$$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$
but I didn't understand how the below formula is generated;
$$\alpha^3 - \beta^3 = (\alpha-\beta)^3+3\alpha\beta(\alpha-\beta)$$
| $(\alpha - \beta)^3 +3\alpha\beta(\alpha - \beta) = \alpha^3 -3\alpha^2\beta + 3\alpha\beta^2 -\beta^3+3\alpha^2\beta - 3\alpha\beta^2 =\alpha^3-\beta^3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1080902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding the residue of $\frac{e^{zt}}{z^2(e^{\pi z}-1)}$ using the laurent series $$f(z)=\frac{e^{zt}}{z^2(e^{\pi z}-1)}\tag{1}$$
I am doing it as follows, but my textbook is convinced that i am wrong:
$ \color{lime}{1.\space Residue\space at\space z\space =0}$
let $\pi z := q$
$e^{q}-1 = q+\frac{q^2}{2}+\frac{q^3}{6}+... | Since:
$$ e^{zt} = 1+zt+\frac{t^2}{2}z^2+O(z^3),$$
$$\frac{z}{e^{\pi z}-1}=\frac{1}{\pi}-\frac{1}{2}z+\frac{\pi}{12}z^2+O(z^3) $$
we have:
$$\begin{eqnarray*}\operatorname{Res}\left(\frac{e^{zt}}{z^2(e^{\pi z}-1)},z=0\right)&=&[z^2]\left(1+zt+\frac{t^2}{2}z^2\right)\cdot\left(\frac{1}{\pi}-\frac{1}{2}z+\frac{\pi}{12}z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1081411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solve $\sin(x)=-\frac{1}{2}$ I have to solve the following equation for $x$
$$\cos(2x)+\sin(x)=0$$
After simplification i got
$\sin(x)=1$ or $\sin(x)=-\frac{1}{2}$
$\Rightarrow x=90^0$
But don't know how to solve for $x$ for $\sin(x)=-\frac{1}{2}$ ?
| You can observe that $-\sin x=\cos(\pi/2+x)$, so your equation becomes
$$
\cos2x=\cos\left(\frac{\pi}{2}+x\right)
$$
which is equivalent to
$$
2x=\frac{\pi}{2}+x+2k\pi
\qquad\text{or}\qquad
2x=-\frac{\pi}{2}-x+2k\pi
$$
and so, after simplifying,
$$
x=\frac{\pi}{2}+2k\pi
\qquad\text{or}\qquad
x=-\frac{\pi}{6}+2k\frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1083330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
find x+y if $ (x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=42 $? Let $ (x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=42 $.
How find $x+y$? I tried a number of ways.
| There isn't just one possible value for $x + y$, which we can see as follows. Let $a = x + \sqrt{x^2 + 1}$ and $b = y + \sqrt{y^2 + 1}$. Then ${1 \over a} = -x + \sqrt{x^2 + 1}$ and ${1 \over b} = -y + \sqrt{y^2 + 1}$. As a result we have
$$x + y = {1 \over 2}\bigg(a - {1 \over a}\bigg) + {1 \over 2}\bigg(b - {1 \over ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Evaluating$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $using residues I need help to solve the next improper integral using complex analysis:
$$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $$
I have problems when I try to find residues for the function $ f = \displaystyle \frac{1}{(z^4+4)^2}$.
This is what... | If you are right, then $\int_{-\infty}^{\infty} \frac{x^2}{1+x^4} dx$ can be easily done by a semi-circle contour, computing the residues at $(1+i)/\sqrt{2}$ and $(-1+i)/\sqrt{2}$. It is easy to check that the integral of $\frac{z^2}{1+z^4}$ on the arc of the semi-circle goes to 0.
$\frac{z^2}{1+z^4}=\frac{z^2}{(z-e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1086458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let $C$ be the curve of intersection of the plane $x+y-z=0$ with ellipsoid $\frac{x^2}4+\frac{y^2}5+\frac{z^2}{25}=1$. Let $C$ be the curve of intersection of the plane $x+y-z=0$ and the ellipsoid $$\frac{x^2}4+\frac{y^2}5+\frac{z^2}{25}=1$$ Find the points on $C$ which are farthest and nearest from the origin
When dea... | Since I am lazy to do it by hands, I give you the sage code.
$###lagrange method
sage:x, y, z, lam , mu= var('x, y, z, lam, mu')
sage:f=x^2+y^2+z^2
sage:g1 = x^2/4+y^2/5+z^2/25-1
sage:g2 = x +y -z
sage:h = f - g1 * lam -g2 * mu
sage:gradh = h.gradient([x, y, z, lam, mu])
sage:critical = solve([gradh[0] == 0,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1087814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Evaluating the sum : $\;\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$ How to evaluate this sum?
$$\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$$
Please give some technique. Binomial not working.
| Some details regarding convergence aside, you have
$$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$
so
$$x e^x = \sum_{k=0}^\infty \frac{x^{k+1}}{k!}$$
and
$$ \int_0^t x e^x \, dx = \sum_{k=0}^\infty \frac{1}{k+2} \frac{t^{k+2}}{k!}.$$
Thus
$$\sum_{k=1}^\infty \frac{1}{k+2} \frac{1}{k!} = \int_0^1 x e^x \, dx - 1.$$
The last... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Prove that $5\mid x\,$ if $\,x, y \gt 1 $ satisfy $2x^2 - 1=y^{15}$ If $x\gt 1$ and $\,y\gt 1,$ with $ x, y \in \mathbb N$ so that $(x,y)$ satisfies the equation $$2x^2-1=y^{15},$$ then prove that $5\mid x$.
$\mod {10}$ gave me just what the last digit of y can be.
| apply fermat, so $y^{15}\equiv y^3 \bmod 5$. So you have $2x^2\equiv y^3+1$. now cubes $\bmod 5$ are $1,3,2,4,0$. This tells us the statement is false, because when $y^3\equiv 1$ we get $2x^2\equiv 2$ and $x$ can be $1$ or $4 \bmod 5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1089842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}$ Good evening everyone,
how can I prove that
$$\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}\;?$$
Well, I know that $\displaystyle\frac{1}{x^4+x^2+1} $ is an even function and the interval $(-\infty,+\infty)$ is sy... | $$\int_{\mathbb{R}}\frac{dx}{x^4+x^2+1}=2\int_{0}^{+\infty}\frac{dx}{x^4+x^2+1} = 2\int_{0}^{1}\frac{dx}{x^4+x^2+1}+2\int_{0}^{1}\frac{x^2\,dx}{x^4+x^2+1}$$
so we just have to find:
$$ I=2\int_{0}^{1}\frac{1+x^2}{1+x^2+x^4}\,dx = 2\int_{0}^{1}\frac{1-x^4}{1-x^6}\,dx.$$
By expanding the integrand function as a geometric... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
} |
What is the solution set of the given homogenous system? Write the solution set of the given homogenous system in parametric vector form:
\begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\
-4x_{1}-4x_{2}-8x_{3} &= 0\\
0x_{1}-3x_{2}-3x_{3} &= 0\\
\end{align}
My attempt:
\begin{align} 2x_{1}+2x_{2}+4x_{3} &= 0\\
-4x_{1}-4x_{2}-8x... | That looks fine so far except that the bottom left $0$ in the very last equation should be a $-1$. The middle equation tells you nothing, so you can eliminate it. You'll end up with a whole line of solutions (all multiples of a single vector).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to proof that $\sum_{i=1}^{2^n} 1/i \ge 1+n/2$ I had troubles trying to prove that for every $n\ge1$
$$\sum_{i=1}^{2^n}\frac1i\ge 1+\frac n2$$
Can you give me a hint about the induction proof or show me in detail how can I prove it? I would appreaciate any help. Thanks!
| $$A=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{2^n}=\\\frac{1}{1}\\+\frac{1}{2}\\+(\frac{1}{3}+\frac{1}{4})\\+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})\\+...\\+(\frac{1}{2^n+1}+\frac{1}{2^n+2}+...+\frac{1}{2^n+2^n})$$now see
$$\frac{1}{1}\\+\frac{1}{2}\\+(\frac{1}{3}+\frac{1}{4})>(\frac{1}{4... | {
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"source": "stackexchange",
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Folding a rectangular paper and finding the area of the triangle so formed. Given a rectangular sheet of paper ABCD such that the lengths of AB and AD are respectively 7 and 3 cms.Suppose B' and D' are two points on AB and AD respectively such that if the paper is folded along B'D' then A falls on A' on the side DC. De... |
Rectangle $ABCD$ is drawn above with vertex $A$ at the origin. Let $DA' = x $ then slope of $AA' = \dfrac{3}{x} $ and therefore, slope of $B'D' = - \dfrac{x}{3} $ and we have that $B'D'$ passes through $( \dfrac{x}{2}, \dfrac{3}{2} )$, therefore, its equation is
$y' = \dfrac{3}{2} - \dfrac{x}{3} (x' - \dfrac{x}{2} ) ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sequence pattern question I have the following question. Let $S_1$ be the sequence of positive integers $1,2,3,4,5 , \ldots$ and define sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to the integers of $S_n$ which are divisible by $n$. I need to find integers $n$ such that the first $n-1$ integers in $S_n$ are $n$.... | Hint: $S_k$ is weakly increasing-it does not decrease because no number can pass another. A prime $p$ is not increased until $S_p \to S_{p+1}$
| {
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"source": "stackexchange",
"question_score": "3",
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Number of solutions to 3-variable sums with restrictions I came across the following problem
1) How many solutions does the equation $x_1+x_2+x_3=8$ have with integers $x_i\ge0$?
There are $9$ possible values for $x_1$. For each of those, there are $9-x_1$ possible values for $x_2$. And for each of those, there is on... | Solving the equation $x_1 + x_2 + x_3 = 8$ involves determining how many ways two plus signs can be placed in a list of eight 1's. For instance, the list
$$1 1 1 + + 1 1 1 1 1$$
corresponds to the solution $x_1 = 3, x_2 = 0, x_3 = 5$, while
$$1 + 1 1 1 1 + 1 1 1$$
corresponds to the solution $x_1 = 1, x_2 = 4, x_3 =... | {
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"source": "stackexchange",
"question_score": "2",
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How to find $ \binom {1}{k} + \binom {2}{k} + \binom{3}{k} + ... + \binom{n}{k} $
Find $$ \binom {1}{k} + \binom{2}{k} + \binom{3}{k} + ... + \binom {n}{k} $$ if $0 \le k \le n$
Any method for solving this problem? I've not achieved anything so far. Thanks in advance!
| No formal answer, but a nice illustration:
The solution can exemplarically be shown by a matrix-multiplication-scheme. The following shows $S \cdot P = X$, where $X$ is simply a shifting of $P$ .
The left-bottom is matrix $S$ which performs the partial summations of the columns of matrix $P$ which is in the right-top-m... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Non-linear differential equation Let $ x \in \mathbb{R} $
Find all solutions of the following differential equation :
$$
y'=\tan(y+x)
$$
| Let $u = y + x$. Then $u' = y' + 1$, so the ODE can be written $$u' = 1 + \tan(u),$$ which is a separable equation. We have $$\int \frac{du}{1 + \tan u} = \int dx,$$ which reduces to $$\frac{1}{2}(u + \log|\sin u + \cos u|) = x + C.$$ To see this, let $$I = \int \frac{du}{1 + \tan u}.$$ Then $$I = \int \left(1 - \frac... | {
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"timestamp": "2023-03-29T00:00:00",
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$ \lim_{n \to \infty} \frac {\sqrt{n+1}+\sqrt{n+2}+...+\sqrt{2n}}{n^{3/2}}$ Evaluate the limit
$$
\lim_{n \to \infty} \frac {\sqrt{n+1}+\sqrt{n+2}+...+\sqrt{2n}}{n^{3/2}}.
$$
Rearranging I can get
$$
\lim_{n \to \infty} \frac {\sqrt{\frac{n+1}{n}}+\sqrt{\frac{n+2}{n}}+...+\sqrt{\frac{2n}{n}}}{n}.
$$
but I do not see ... | Using Stolz-Cesaro:
$$\eqalign{
\lim_{n\to\infty}\frac{\sqrt{n+1}+\cdots+\sqrt{2n}}{\sqrt{n^3}}
& =
\lim_{n\to\infty}\frac{(\sqrt{(n+1)+1}+\cdots+\sqrt{2(n+1))}-(\sqrt{n+1}+\cdots+\sqrt{2n})}{\sqrt{(n+1)^3}-\sqrt{n^3}}\cr
& =
\lim_{n\to\infty}\frac{\sqrt{2n+1}+\sqrt{2(n+1)}-\sqrt{n+1}}{\sqrt{(n+1)^3}-\sqrt{n^3}}\cr
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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integration of $\large\int \frac{u^2}{(1-u^2)^2}$ $ du$ Is there a way to integrate $$\large\int \frac{u^2}{(1-u^2)^2} du$$ without using partial fraction decomposition?
| Hint:
Let $f(u) = u$ and $g(u) = 1-u^2$. Hence,
$$ \left(\frac{f}{g}\right)'(u) = \frac{u^2+1}{(1-u^2)^2} = \frac{2u^2}{(1-u^2)^2} + \frac{1-u^2}{(1-u^2)^2} = \frac{2u^2}{(1-u^2)^2} + \frac{1}{2} \left(\frac{1}{1-u} + \frac{1}{1+u}\right).$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving a differential equation $F-y'F_{y'}=C$, with $F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}}$ If $$F= F(y,y')= \frac{1+2y'^2}{3y^3\sqrt{1+y'^2}},$$
where $y=y(x)$ and $y'= y'(x)=\frac{dy}{dx}$, then how to solve the differential equation:
$$F-y'F_{y'}=C, $$
that is:
$$F(y,y')-\frac{dy}{dx}\frac{\partial F}{\part... | We have that
$$\begin{align} \\
F(y,y') &= \frac{1 + 2y'^{2}}{3y^{3} \sqrt {1+y'^{2}}} \\
&= \frac{1}{3y^{3}}\bigg[(1 + 2y'^{2})(1 + y'^{2})^{\frac{-1}{2}}\bigg]
\end{align} $$
As there is no explicit $x$ dependence, we can find a first integral of the form
$$\begin{align} \\
y'\frac{\partial F}{\partial y'} - F &= y'\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101180",
"timestamp": "2023-03-29T00:00:00",
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Evaluating$\int\frac{1}{(x^2-1)^2}$ This is the integral:
$\int\frac{1}{(x^2-1)^2}$
I have tried several ways to solve this but I always end up that last parameter equals 1 and all others equals 0 so I end up where I started. Examples over the internet with similar fraction have more than $1$ in the numerator which mak... | Since:
$$\frac{1}{x^2-1}=\frac{1}{(x-1)(x+1)}=\frac{1}{2}\left(\frac{1}{x-1}-\frac{1}{x+1}\right),$$
then:
$$\frac{1}{(x^2-1)^2}=\frac{1}{4}\left(\frac{1}{(x-1)^2}-\frac{2}{(x-1)(x+1)}+\frac{1}{(x+1)^2}\right)$$
or:
$$\frac{1}{(x^2-1)^2}=\frac{1}{4}\left(\frac{1}{(x-1)^2}-\frac{1}{x-1}+\frac{1}{x+1}+\frac{1}{(x+1)^2}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102071",
"timestamp": "2023-03-29T00:00:00",
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Evaluating $\int\frac{\sqrt{x^2-1}}x\mathrm dx$ How can one evaluate the integral
$$\int\frac{\sqrt{x^2-1}}x\mathrm dx$$?
I tried substituting $x = \cosh t$ but got stuck at
$$\int\frac{\sinh^2t}{\cosh t}\mathrm dt$$
Any hints?
| $$
\begin{aligned}\int \frac{\sqrt{x^{2}-1}}{x} d x =& \int \frac{x^{2}-1}{x \sqrt{x^{2}-1}} d x \\
=& \int \frac{x^{2}-1}{x^{2}} d\left(\sqrt{x^{2}-1}\right) \\
=& \int\left(1-\frac{1}{x^{2}}\right) d\left(\sqrt{x^{2}-1}\right) \\
=& \sqrt{x^{2}-1}-\int \frac{d\left(\sqrt{x^{2}-1}\right)}{\left(\sqrt{x^{2}-1}\right)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Pythagorean triples So I am given that $65 = 1^2 + 8^2 = 7^2 + 4^2$ , how can I use this observation to find two Pythagorean triangles with hypotenuse of 65.
I know that I need to find integers $a$ and $b$ such that $a^2 + b^2 = 65^2$, but I don't understand how to derive them from that observation.
Here is my attempt.... | As I said we need to solve a system of equations.
The system of equations:
$$\left\{\begin{aligned}&x^2+y^2=z^2\\&q^2+t^2=z^2\end{aligned}\right.$$
the solutions have the form: $$x=4p^4-s^4$$ $$y=4p^2s^2$$ $$q=4ps(2p^2-s^2)$$ $$t=4p^4-8p^2s^2+s^4$$ $$z=4p^4+s^4$$
$p,s,k$ - integers.
Formulas you... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combinatorial Proof For Counting All Binary Strings The Question
Provide a combinatorial proof for the following:
For $n \ge 1$,
$$2^n = \binom{n+1}1+\binom{n+1}3+\ldots+\begin{cases}
\binom{n+1}n,&\text{if }n\text{ is odd}\\\\
\binom{n+1}{n+1},&\text{if }n\text{ is even}
\end{cases}$$
My Work
Parts a,b of the question... | Your $n=4$ line should read
$$2^4=\binom51+\binom53+\binom55=5+10+1=16\;.$$
The upper number in each of the binomial coefficients is $n+1$, which is $5$ in this case, and the lower number runs through all of the odd numbers less than or equal to $n+1$.
HINT:
$$\binom{n+1}1+\binom{n+1}3+\ldots+\begin{cases}
\binom{n+1}... | {
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"question_score": "3",
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Show that if $a$ is an integer, then 3 divides $a^3 - a $ Show that if $a$ is an integer, then 3 divides $a^3 - a $
we can write, where $k$ is an integer;
$a^3 - a = 3k $
$a(a^2 - 1) = 3k $
Now if $a = k$ then
$a^2 -1 = 3$ and $a= \pm2 $ so $ a^3 - a = 24 = 3 \times 8$
If $ a $ is not equal to $k$;
then
$a(a^2 - 1) = a... | $a^3-a=(a-1)a(a+1)$ is always true and therefore is always the product of three consecutive integers.
Another way is to see that $a$ can be congruent to $0$, $1$ or $2$ $mod 3$. In each case we get that $a^3-a$ is congruent to $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Laurent series expansion, can one simplify this? I have to expand $f(z)=\frac{z-1}{(z^2+1)z}$ in an annulus $R(i,1,2)$.
$$f(z)=\frac{1}{z-i}\frac{1}{z+i}-\frac{1}{z-i}\Big(\frac{i}{z+i}-\frac{i}{z}\Big)$$
$$\frac{1}{z-i}\frac{1}{z+i}=\frac{1}{z-i}\cdot\frac{1}{2i}\cdot\frac{1}{1-(-(z-i)/2i)}=\sum_{n=0}^{\infty}(-1)^n... | Hi this series as generalitation of Laurent Series in three poles
$$\sum _{k=-1}^n \left(\frac{1}{2}+\frac{i}{2}\right) \left((-i)^k-i i^k\right) x^k+\frac{(-1)^{n-1} \left(\left(\frac{1}{2}-\frac{i}{2}\right) x^{n+1}\right)}{x-i}+\frac{(-1)^{n-1} \left(\left(\frac{1}{2}+\frac{i}{2}\right) x^{n+1}\right)}{x+i}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109459",
"timestamp": "2023-03-29T00:00:00",
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$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$ $$I=\int \frac{\cos^3(x)}{\sqrt{\sin^7(x)}}\,dx$$
I tried to write it as
$$I=\int \sqrt{\frac{\cos^6(x)}{\sin^7(x)}}\,dx$$
And $$I=\int \sqrt{\frac{1}{\tan^6(x)\sin(x)}}\,dx$$ but it seems to go nowhere , how can I manipulate it so that its becomes solvable?
| Here are the steps
$$\int \frac{\cos^3 x}{\sqrt{\sin^7 x}}dx= \int \frac{(1-\sin^2 x)(\cos x)}{\sqrt{\sin^7 x}}dx $$
Let $u=\sin x$, then $du=\cos x\ dx$. So now
$$ \int \frac{1-u^2}{\sqrt{u^7}}du = \int \frac{1}{\sqrt{u^7}}-\frac{u^2}{\sqrt{u^7}}du $$
$$= \int u^{-\frac72}du-\int u^{-\frac32}du $$
$$= \frac{u^{-\frac7... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit $\lim_\limits{x\to0} \frac{\ln\left(x+\sqrt{1+x^2}\right)-x}{\tan^3(x)}$ Evaluate the given limit:
$$\lim_{x\to0} \frac{\ln\left(x+\sqrt{1+x^2}\right)-x}{\tan^3(x)} .$$
I've tried to evaluate it but I always get stuck... Obviously I need L'Hôpital's Rule here, but still get confused on the way. May someone show m... | Recall that
$$\text{arsinh } x := \ln\left(x + \sqrt{1 + x^2}\right),$$
and that it has a nice Taylor series expansion:
$$\text{arsinh } x \sim x - \frac{1}{2} \cdot \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} - \cdots$$
(it's not too hard to write the coefficients in a closed form, but we only need the f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Commutative matrices Knowing that $AB=BA$, find the matrices that commute with the matrix
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
I have assumed that multiplying matrix
$\begin{pmatrix}
a & b \\
c & d \end{pmatrix}
$ by the first one should be equal to multiplying the first one by $\begin{pmatrix}
... | You should be left with a little more than two equations. If $$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$$
and $$B=\begin{bmatrix}a&b\\c&d\end{bmatrix},$$
then $$AB=\begin{bmatrix}a+2c & b+2d\\ 3a+4c & 3b+4d\end{bmatrix}$$
and $$BA=\begin{bmatrix}a+3b & 2a+4b\\ c+3d & 2c+4d\end{bmatrix}$$
which gives you $4$ equations.
| {
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Find all positive integers $n$ for some given condition. Find all positive integers $n>1$ such that $n^2$ divides $2^n+1$
I found that $n$ is of the form $6k+3$.
| The only answer is $n=3$
(1) Since $n$ divides $2^n + 1$, $n$ is odd. Let $p$ be the smallest prime divisor of $n$
(2) Let $a$ be the smallest positive integer such that $2^a \equiv -1 \pmod p$. $a$ must exist since $2^n \equiv -1 \pmod p$
(3) Let $b$ be the smallest positive integer such that $2^b \equiv 1 \pmod p... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate this infinite sum $$s= \sum_{n=1}^\infty \frac{n+3}{(2^n)(n+1)(n+2)}$$
Any method to calculate this type of infinite sums?
| Using Partial Fraction Decomposition,
$$\text{let }\frac{n+3}{(n+1)(n+2)}=\frac A{n+1}+\frac B{n+2}$$
$$n+3=n(A+B)+2A+B\implies A+B=1,2A+B=3\implies A=2, B=-1$$
$$\implies\frac{n+3}{(n+1)(n+2)}=\frac 2{n+1}-\frac1{n+2}$$
$$\implies\frac{n+3}{2^n(n+1)(n+2)}=\frac{(1/2)^{n-1}}{n+1}-\frac{(1/2)^n}{n+2}$$
If $T_m=\dfrac{(1... | {
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Prove that if $({x+\sqrt{x^2+1}})({y+\sqrt{y^2+1}})=1$ then $x+y=0$ Let
$$\left({x+\sqrt{x^2+1}}\right)\left({y+\sqrt{y^2+1}}\right)=1$$
Prove that $x+y=0$.
This is my solution:
Let
$$a=x+\sqrt{x^2+1}$$
and
$$b=y+\sqrt{y^2+1}$$
Then $x=\dfrac{a^2-1}{2a}$ and $y=\dfrac{b^2-1}{2b}$. Now $ab=1\implies b=\dfrac1a$. Then ... | Note
$$y+\sqrt{y^2+1}=\sqrt{x^2+1}-x\tag{1}$$
$$x+\sqrt{x^2+1}=\sqrt{y^2+1}-y\tag{2}$$
$(1)+(2)$
$$\Longrightarrow x+y=-(x+y)$$
$$\Longrightarrow x+y=0$$
| {
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"url": "https://math.stackexchange.com/questions/1118742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How to integrate $1/\sqrt{(1+x^2)^3}$? Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem.
Is anyone able to assist me with a walk through of atleast the start if not all of the steps to solving this equa... | $$\int\frac{dx}{\sqrt{(1+x^2)^3}}$$
make a trigonometric substituition
$$
x=\tan\theta\\
dx=\sec^2\theta d\theta\\
1+x^2=\sec^2\theta\\
\int\frac{\sec^2\theta}{\sqrt{(\sec^2\theta)^3}}d\theta$$
if $\theta\in(0|\frac{\pi}{2})\cup(\frac{3\pi}{2}|2\pi)$ then $\cos\theta>0$ and then
$$\begin{align}
\tan\theta&=x\\
\sec\the... | {
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"timestamp": "2023-03-29T00:00:00",
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How to integrate $(x^2 - y^2) / (x^2 + y^2)^2$ How do I integrate
$$\int \int \frac{(x^2 - y^2)}{(x^2 + y^2)^2} dx dy?$$
The WolframAlpha page gives
$$
c_1 + c_2 + \tan^{-1}(x/y).
$$
And I kind of specifically need
$$
\int_{0}^{x} \frac{(x^2 - y^2)}{(x^2 + y^2)^2} dy.
$$
Note
*
*I want to know integration techniqu... | By the quotient rule $$\frac{\partial}{\partial y}\left(\frac{y}{x^2 + y^2}\right) = \frac{x^2 - y^2}{(x^2 + y^2)^2}.$$
Therefore
$$\int_0^x \frac{x^2 - y^2}{(x^2 + y^2)^2}\, dy = \int_0^x \frac{\partial}{\partial y}\left(\frac{y}{x^2 + y^2}\right)\, dy = \frac{y}{x^2 + y^2}\bigg|_{y = 0}^{y = x} = \frac{1}{2x}.$$
| {
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"answer_id": 0
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Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$
Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$
Abel's/Dirichlet's tests cannot be applied here.
I guess it's something more tricky involving integration maybe (?)
| First write
$$\sum_{n = 1}^\infty \frac{(-1)^n\cos^2 n}{n} = \sum_{n = 1}^\infty \left(\frac{(-1)^n}{2n} + \frac{(-e^{2i})^n}{4n} + \frac{(-e^{-2i})^n}{4n}\right)$$
using the identities $$\cos^2 x = \frac{(e^{ix} + e^{-ix})^2}{4} = \frac{1}{2} + \frac{e^{2ix}}{4} + \frac{e^{-2ix}}{4}.$$ The series $\sum_{n = 1}^\infty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1120497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Integral $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx$, where $0I am trying to evaluate the following integral:
$$\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx,$$
where $0<c<1$.
I can't really think of a way to find it so please give me a hint.
| Express the integrand as a series:
$$\begin{align}I(c) &= \int_{-\pi/2}^{\pi/2} dx \, \log{(1+c \sin{x})} \\ &= \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} c^k \int_{-\pi/2}^{\pi/2} dx \, \sin^k{x} \\ &= - \frac{\pi}{2}\sum_{k=1}^{\infty} \frac{c^{2 k}}{k} \frac1{2^{2 k}} \binom{2 k}{k} \end{align} $$
$$I'(c) = -\pi \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
To show two matrices are conjugate to each other Given two matrices A and B
$$
A =
\begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad
B =
\begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
$$
I have checked basic properties such as trace, charatersitic polynomial and are same .
... | Compute the eigenvalues of $A$ and $B$. These are of course $1, 1, 3$.
Compute eigenvectors of $A$. These are
$$
\begin{bmatrix}-2\\1\\0\end{bmatrix},
\begin{bmatrix}0\\0\\1\end{bmatrix},
\begin{bmatrix}0\\1\\1\end{bmatrix},
$$
where the first two are relative to $1$, and the third to $3$.
Simlarly for $B$ one has
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1121997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Algebra QF $f(x)=x^2-18x-4$ I have the problem to find the zero of the function:
$$f(x)=x^2-18x-4$$
I have it mostly worked out as $a=1, b= -18, c= -4$
worked out I have:
$$\frac{18\pm \sqrt{340}}{2}$$
I know the answers are
$$9+ \sqrt{85}\\
9- \sqrt{85}$$
The question is when working the problem out I end up with:
$$... | Note that:
$$\frac{18\pm\sqrt{340}}{2}=\frac{18\pm\sqrt{4\cdot85}}{2}=\frac{18\pm\sqrt{4}\cdot\sqrt{85}}{2}=\frac{18\pm2\sqrt{85}}{2}=\frac{2\cdot(9\pm\sqrt{85})}{2}=9\pm\sqrt{85}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Recurrence Relations Closed Form So, the question is to derive the closed form solution to the recurrence relation $$T(n) = 3T(n-1) + 5,\hspace{5mm} T(0) = 0.$$
$\begin{align}T(n) &= 3T(n-1)+5
\\&= 3(3T(n-2)+5)+5
\\&= 3(3(3T(n-3)+5)+5)+5\end{align}$
I'm just struggling to go from here. I understand that it's $... | Calculating the first few values: $0,5,20,65,200$ , dividing by $5$ this is $0,1,4,13,40$ , multiplying by $2$ we get $0,2,8,26,80$ adding $1$ we get $1,3,9,27,81$. Otherwise you can use generating functions:
Let $A=\sum\limits_{n=0}^\infty t(n)x^n$.
Then $A=3xA+5\sum\limits_{n=1}^\infty x^n$ (since $f(0)=0$).
Then $(1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What general function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3…? Look at this sequence:
2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3...
It is defined as follows:
$$f(n)=\begin{cases}
3 &\text{if $n \bmod 7=6,0$}\\
2 &\text{otherwise}\\
\end{cases}$$
David found a good representa... | If I understand what you are asking, we can define
$$
\newcommand{\flfrac}[2]{\left\lfloor\frac{#1}{#2}\right\rfloor}
f(n)=b+(a-b)\left(\flfrac{n+d}c-\flfrac{n-1}c\right)\tag{1}
$$
then
$$
f(n)=\left\{\begin{array}{}
a&\text{if }n\equiv c-d\dots c\pmod{c}\\
b&\text{if }n\equiv1\dots c-d-1\pmod{c}
\end{array}\right.\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1125017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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finding $\int\frac{1}{(t^2+25)^2} dt$ without trig substitution Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate
$$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$
without using trig substitution (or partial fractions over the complex num... | $$
\begin{aligned}
\int \frac{1}{\left(t^{2}+25\right)^{2}} d t &=-\frac{1}{2} \int \frac{1}{t} d\left(\frac{1}{t^{2}+25}\right) \\
&=-\frac{1}{2 t\left(t^{2}+25\right)}-\frac{1}{2}\left(\frac{1}{t^{2}\left(t^{2}+25\right)} d t\right.\\
&=-\frac{1}{2 t\left(t^{2}+25\right)}-\frac{1}{50} \int\left(\frac{1}{t^{2}}-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function? Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
| x^3+bx^2+cx+d = 0 has 3 solutions 3 and 2+i are given. third solution z is not.
this polynomial can decomposed as (x-3)(x-2-i)(x-z)=0 from here follows
1. 3*(2+i)*z = -d
2. 3z + 3(2+i) + z(2+i) = c
3. 3 + 2+i + z = -b
these admits a solution:
b = -5 - i - z;
c = 6 + 3 i + 5 z + i z,
d = -3 z(2 + i)
in order b to be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that a recurrence relation (containing two recurrences) equals a given closed-form formula. Prove that $a_n = 3a_{n-1} - 2a_{n-2} = 2^n + 1$ , for all $n \in \mathbb{N}$ , and $a_1 = 3$ , $a_2 = 5$ , and $n \geq 3$
Basis:
$a_1 = 2^1 + 1 = 2 + 1 = 3$ $\checkmark$
$a_2 = 2^2 + 1 = 4 + 1 = 5$ $\checkmark$
Inductive ... | $$3\cdot 2^k - 2\cdot 2^{k-1}= 3\cdot 2^k -2^k=2^k(3-1)=2^k\cdot2=2^{k+1}$$
Notice that $$ 3\cdot 2^k = 2^k + 2^k + 2^k$$
While
$$ 6^k=(3 \cdot 2)^k=3^k\cdot 2^k=2^k + 2^k ....... + 2^k$$
The last line shows $2^k$ being added $3^k$. There is a way to show this better in latex (a bracket under the ellipses) but I don't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1127212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$. Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$.
My attempt:
$(a+1)(a-1)+(b+1)(b-1)=c^2+1$
This form didn't help so I thought of $\mod 3$, but that didn't help either. Please help. Thank you.... | Put $a=6k^2-2$, $b=6k$ and $c=6k^2+1$. Then
$$\left(6k^2-2\right)^2+(6k)^2=\left(6k^2+1\right)^2+3.$$
Behind these solutions is the observation that $$\left(\frac{x+y}{2}\right)^2-\left(\frac{x-y}{2}\right)^2=xy$$ We can re-write the given equation as $c^2-a^2=b^2-3$ and choose $b$ such that $b^2-3$ is the product of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1127860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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$f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$ For all $x,y\in\mathbb{R}$. also $f : \mathbb{R} → \mathbb{R}$ and $x+y\not=0$.
My attempt:
I restated it as
$a[x^2 y^2 (\frac{x}{y}+\frac{y}{x}-\frac{1}{y^2}-\frac{1}{x^2})] +
b[xy(x+y-\frac{1}{y}-\frac{1}{x})] + c [x+y-2]=0$
because of $f(xy)... | Let $y=1$
$f(x)(x+1)=f(x)+f(1)$ Hence $$f(x)=\frac{f(1)}{x}$$
You can verify that : $$f(xy)=\frac{f(1)}{xy}=\frac{\frac{f(1)}{x}+\frac{f(1)}{y}}{x+y}$$
As Peter Taylor explained in the comment :
$$f(0)=f(1)+f(0)$$
So if you want $f$ to be defined on $\mathbb R$, you need to have $f(1)=0$. Hence $$\forall x\neq 0\quad f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Surface Area by Integration $$2\pi\int_{3}^6\left(\frac{1}{3}x^\frac{3}{2}-x^\frac{1}{2}\right)\left(1+\left(\frac{1}{2}x^\frac{1}{2}-\frac{1}{2}x^\frac{-1}{2}\right)^2\right)^\frac{1}{2}dx$$
I've managed to simplify this down to the equation below (not sure if it'll help), but I still can't integrate it.
Please help.... | HINT
I would say
$x^4-4x^3-2x^2+12x+9=(x^2-2x-3)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving the inequality $(x^2+3)/x\le 4$ This is the inequality
$$\left(\frac {x^2 + 3}{x}\right) \le 4 $$
This is how I solve it
The $x$ in the left side is canceled and $4x$ is subtracted from both sides.
$$\not{x} \left (\frac {x^2+3} {\not{x}}\right) \le 4x $$
$$ x^2+3 - 4x \le 4x - 4x $$
$$x^2 -4x + 3 \le 0 $$
Then... | Never multiply an inequality by a variable since when multiplying by a negative number you have to change the sign.This is how this kind of problems should be solved
$$\frac{x^2+3}{x}\leq 4\\\frac{x^2+3}{x}-4\leq 0\\\frac{x^2-4x+3}{x}\leq 0\\\frac{(x-3)(x-1)}{x}\leq 0$$
Now you should spit it into cases
*
*$x<0$ The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Hockey pucks and parameters There is one hockey puck with a diameter of $3$ inches. The puck is spinning around its center at a speed of $3$ counterclockwise rotations per second. At the center, the puck is traveling at a speed of $24$ inches per second at an angle of $45^\circ$ to the positive $x$-axis.
(a) At time $... | (a) The hockey puck is traveling along the line $y=x$, traveling 24 inches every second. Since the puck moves 24 inches when $t=1$ and the puck travels at a constant pace, we can graph:
When $t=1$, $x = y = 12\sqrt{2}$. Therefore, the center of the puck at time $t$ is at:
\begin{align*}
x &= 12\sqrt{2}t \\
y &= 12\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1133788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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exercise: ordinary differential equations I am struggling with an exercise. Can you please give me a hint?
Exercise:
Show that the solution curves of the differential equation:
$\frac{dy}{dx}=-\frac{y(2x^3-y^3)}{x(2y^3-x^3)}$, are of the form $x^3+y^3=3Cxy$.
I tried the substitution $u=y/x \rightarrow y=xu, \frac{dy}{d... | $\bf hint:$ so you need to integrate $\int \frac{2u^3 - 1}{u^4 + u}$ we can do this by using partial fraction. here is how it goes
$$ \frac{2u^3 - 1}{u^4 + u} =
\frac{1}{2}\left(\frac{4u^3 +1}{u^4 + u} - \frac{3}{u(u+1)(u^2-u+1)}\right)$$
now you need to find the constants $A, B, C$ and $D$ so that $$ \frac{3}{u(u+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Partial Fraction Decomposition Clarification I'm just looking for some overall clarification for the following cases. Now, to the extent of my knowledge, the following examples of partial fractions would be split up in the following way:
\begin{align}
\frac{1}{x^2+3x-4}&=\frac{1}{\left(x+4\right)\left(x-1\right)}=\frac... | $$
x^3+x^2+x+1 = x^2(x+1) + 1(x+1) = (x^2+1)(x+1)
$$
So
$$
\frac{\cdots\cdots\cdots}{x^3+x^2+x+1} = \frac{Cx+D}{x^2+1} + \frac{E}{x+1}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is there different results when calculating a double sum? Consider:
$$
\begin{matrix}
-1 & 0 & 0 & 0& 0& \ldots\\
1/2 & -1 & 0 & 0& 0& \ldots\\
1/4 & 1/2 & -1 & 0& 0& \ldots\\
1/8 & 1/4 & 1/2 & -1& 0& \ldots\\
1/16&1/8 & 1/4 & 1/2 & -1& \ldots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{matrix}
$$
When I calcu... | The sum is not absolutely convergent (http://en.wikipedia.org/wiki/Convergent_series) so we can make the sum equal whatever we want.
With divergent series the way that we order the series may change the sum. As @DuAravis said the series can be written as
\begin{align*}
S_n &=(-1) +(\frac{1}{2} -1) +(\frac{1}{4}+\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1137509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving $1+\frac{4}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{4}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\frac{4}{10^2}+\frac{1}{11^2}+\cdots=\frac{\pi ^2}{4}$ Proving $$1+\frac{4}{2^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{4}{6^2}+\frac{1}{7^2}+\frac{1}{9^2}+\frac{4}{10^2}+\frac{1}{11^2}+\cdots=\frac{\pi ^2}{4}$$
Firstly, I thought... | This sum is explicitly:
$$\sum_{n=1}^\infty\frac{1}{n^2}+3\frac{1}{(2n)^2}-4\frac{1}{(4n)^2}$$
You can check for yourself that this eliminates $n$'s which are a multiple of four, while multiplying by four terms with even $n$ but which are not a multiple of four. In other terms:
$$\sum_{n=1}^\infty\frac{1}{n^2}\left(1+\... | {
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"url": "https://math.stackexchange.com/questions/1137827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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How to solve $y′′′+y'=2-\sin(x)$ I have tried to solve this but with no luck.
So far I just get,
$$y_p(x) = A \sin x + B \cos x \\
y'_p(x) = A \cos x - B \sin x \\
y''_p(x) =-A \sin x - B \cos x \\
y'''_p(x) =-A \cos x + B \sin x $$
$$y'''+ y' = -A \cos x + B \sin x + A \cos x - B \sin x \\ \\
= \cos... | here is a way to why $x$'s are added. we will find a particular solution to $$y'' + y = \sin x \tag 1$$ the equation is forced by a solution of the homogeneous problem(called the resonant case). first you look at a more general problem of the solution of
$$y'' + y = \sin kx$$ a particular solution is $$y_p = \frac{\s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How to compute $\int_0^{\infty} \ln (1 + e^{-x})\, dx$ and $\int_0^{\infty} \ln (1 - e^{-x})\, dx$? Bierenes de Haan's book (page 377) shows that $\int_0^{\infty} \ln (1 + e^{-x})\, dx = \frac{\pi^2}{12}$, and $\int_0^{\infty} \ln (1 - e^{-x})\, dx = -\frac{\pi^2}{6}$. Anybody know how to compute them? Thanks.
| Using the Maclaurin series for $\ln(1 + x)$, we have
\begin{align}\int_0^\infty\ln(1 + e^{-x})\, dx &= \int_0^\infty \sum_{n = 1}^\infty (-1)^{n-1}\frac{e^{-nx}}{n}\, dx \\
&= \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{n}\int_0^\infty e^{-nx}\, dx\\
& = \sum_{n = 1}^\infty \frac{(-1)^{n-1}}{n}\cdot\frac{1}{n}\\
& = \sum_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1141696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Derive a Recurrence Could really use some help with this.
For an integer $m \geq 1$ and $n \geq 1$, consider $m$ horizontal lines and $n$ non-horizontal lines, such that no two of the non-horizontal lines are parallel and no three of the $m+n$ lines intersect in one single point. These lines divide the plane into regio... | Think about what happens each time you add a line of each type. If you add a horizontal line, it will intersect each of the $n$ non-horizontal lines, but none of the $m$ horizontal lines. On the other hand, if you add a non-horizontal line, it will intersect all of the existing $m + n$ lines. After drawing a couple ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's wrong with how I calculated the inverse of this matrix? $\left( \begin{array}{ccc}
-1 & -1 & 2 & | & 1 & 0 & 0\\
2 & 0 & 0 & | & 0 & 1 & 0\\
2 & 2 & 0 & | & 0 & 0 & 1\end{array} \right) $ is the original matrix. Now, add 2*R1 to R2 and R3.
$\left( \begin{array}{ccc}
-1 & -1 & 2 & | & 1 & 0 & 0\\
0 & -2 & 4 & | &... | It seems you made a small computational error in rewriting the right hand inverse matrix after step 3. You gotta be really careful about that,it's like a complicated integration-it can really shaft you on an exam or important homework problem. If a row reduction to yield an inverse matrix looks wrong to you but you're ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1145976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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finding the minimum value of $\frac{x^4+x^2+1}{x^2+x+1}$ given $f(x)=\frac{x^4+x^2+1}{x^2+x+1}$.
Need to find the min value of $f(x)$.
I know it can be easily done by polynomial division but my question is if there's another way
(more elegant maybe) to find the min?
About my way: $f(x)=\frac{x^4+x^2+1}{x^2+x+1}=x^2-x+... | $$x^2-x+1=\frac{4x^2-4x+4}4=\frac{(2x-1)^2+3}4\ge\frac34$$
The equality occurs if $2x-1=0\iff x=\dfrac12$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$ Having the following inequality
$$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$
To prove it for all natural numbers is it enough to show that:
$\frac{1}{(n+1)^2}-\frac{1}{n^2} <2$ or $\frac{1}{(n+1)^2}... | As noted, induction is a more difficult way to prove this. Here it is.
Claim:
$$
\frac{1}{1^2}+\frac{1}{2^2}+\dots+\frac{1}{n^2} < 2-\frac{1}{n}
$$
for $n=2,3,4,\cdots$. First, when $n=2$ we have
$$
\frac{1}{1}+\frac{1}{4} = \frac{5}{4} < \frac{3}{2} = 2-\frac{1}{2}
$$
which is correct.
Now, suppose $n \ge 2$ and
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1150388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
Integrate $ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3 \phi}{\left(2\cos 2 \phi+ 3\right)\sqrt{\cos 2 \phi}}\right)d\phi$ Evaluate the integral:
$$\displaystyle \int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3 \phi}{\left(2\cos 2 \phi+ 3\right)\sqrt{\cos 2 \phi}}\right)d\phi$$
I have no clue ... | By replacing $\phi$ with $\arctan(t)$, then using integration by parts, we have:
$$ I = \int_{0}^{1}\frac{1}{1+t^2}\,\arctan\left(\frac{\sqrt{2}(1-3t^2)}{(5+t^2)\sqrt{1-t^2}}\right)\,dt =\frac{\pi^2}{8}-\int_{0}^{1}\frac{3\sqrt{2}\, t \arctan(t)}{(3-t^2)\sqrt{1-t^2}}\,dt.$$
Now comes the magic. Since:
$$\int \frac{3\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1151817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 1,
"answer_id": 0
} |
how to evaulate: $\lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)} $ How do I evaluate: $\lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)} $ ?
according to Taylor's series, I did like this:
$$\lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)}=\lim \limits_{x \to 0} \frac{x^5 - \f... | We have
\begin{align}
&\lim_{x\to 0} \frac{x^5 - \frac{x^{10}}{2} + O(x^{15})}{(x^3 + \frac{x^6}{2!} + O(x^9))(x^2 - \frac{x^6}{3!} + O(x^{10}))}\\
&= \lim_{x\to 0}\frac{1 + O(x^5)}{(1 + O(x^3))(1 + O(x))}\\\\
&= \frac{1}{(1)(1)}\\
&= 1.
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Implicit derivative switching $y$ and $x$ Regard $y$ as the independent variable and $x$ as the dependent variable and use implicit differentiation to find $dx/dy$ if
$y \sec x = 4x \tan y$
I got $(\sec x-4x\sec^2 y)/4\tan y-y\sec \tan x$ but it was wrong.
| You almost got it.
$\begin{align}
y\sec x & = 4x \tan y
\\[1ex]
y' \sec x + y \tan x \sec x & = 4 \tan y + 4 x y' \sec ^2 y & \text{implicit derivation using the chain rule}
\\[1ex]
y' (\sec x - 4 x \sec ^2 y) & = 4 \tan y - y \tan x \sec x & \text{associate like elements}
\\[1ex]
\frac{\mathrm d y}{\mathrm d x} & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.
Let $a$, $b$ and $c$ be the three sides of a triangle.
Show that $$\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3\,.$$
A full expanding results in:
$$\sum_{cyc}a(a+b-c)(a+c-b)\g... | $a, b, c$ are sides of a triangle iff there exists positive reals $x, y, z$ s.t. $a=x+y, b=y+z, c = z+x$. In terms of these variables, the inequality is
$$\sum_{cyc} \frac{a}{b+c-a} = \sum_{cyc} \frac{x+y}{2z} \ge 3$$
Now the last is easy to show with AM-GM of all $6$ terms.
$$\sum_{cyc} \frac{x+y}{2z} = \frac12\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 0
} |
Proof of a limit of a sequence I want to prove that $$\lim_{n\to\infty} \frac{2n^2+1}{n^2+3n} = 2.$$
Is the following proof valid?
Proof
$\left|\frac{2n^2+1}{n^2+3n} - 2\right|=\left|\frac{1-6n}{n^2+3n}\right| =\frac{6n-1}{n(n+3)} $ (because $n \in \mathbb N^+)$. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... | From $\lim_{n\to\infty} \dfrac{2n^2+1}{n^2+3n}$, divide top and bottom by $n^2$ to get:
$\lim_{n\to\infty} \dfrac{2+1/n^2}{1+3/n}$, which, as $n\to\infty$ becomes
$\dfrac{2+0}{1+0}=\dfrac{2}{1}=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
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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.