Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$? I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
| Since $x := \cos \frac{2 \pi}{5} = \frac{z + z^{-1}}{2}$ where $z:=e^{\frac{2 i \pi}{5}}$, and $1+z+z^2+z^3+z^4=0$ (for $z^5=1$ and $z \neq 1$), $x^2+\frac{x}{2}-\frac{1}{4}=0$, and voilà.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/7695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 11,
"answer_id": 0
} |
The Basel problem As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it w... | I'll post the one I know since it is Euler's, and is quite easy and stays in $\mathbb{R}$. (I'm guessing Euler didn't have tools like residues back then).
Let
$$s = {\sin ^{ - 1}}x$$
Then
$$\int\limits_0^{\frac{\pi }{2}} {sds} = \frac{{{\pi ^2}}}{8}$$
But then
$$\int\limits_0^1 {\frac{{{{\sin }^{ - 1}}x}}{{\sqrt {1 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "814",
"answer_count": 48,
"answer_id": 2
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Proving an identity involving terms in arithmetic progression. If $a_1,\ldots,a_n$ are in arithmetic progression and $a_i\gt 0$ for all $i$, then how to prove the following two identities:
$ (1)\large \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \frac{1}{\sqrt{a_2} + \sqrt{a_3}} + \cdots + \frac{1}{\sqrt{a_{n-1}} + \sqrt{a_n}... | Both identities can be proved quite easily with inductions. Let $d$ be the common difference, i.e. $d=a_{n+1}-a_n$.
*
*Use induction. So we need to prove
$$ \frac{n-2}{\sqrt{a_1}+\sqrt{a_ {n-1}}}+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_ n}}=\frac{n-1}{\sqrt{a_1}+\sqrt{a_ n}}.$$
Rationalize the denominators and substitute $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/10452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Find the value of $\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ}) $ How to find the value of
$$\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ})$$
manually ?
| Like Trigonometry Simplification,
$$\frac{2\sin60^\circ\cdot\cos20^\circ-2(2\sin20^\circ\cos20^\circ)}{\sin20^\circ}$$
Using Werner Formula we get,
$$\frac{\sin80^\circ+\sin40^\circ-2\sin40^\circ}{\sin20^\circ}$$
Using Prosthaphaeresis Formula, $\sin80^\circ-\sin40^\circ=2\sin20^\circ\cos60^\circ$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/10661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
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How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$
how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$
Thank you
| Perhaps a sixth way...
$$\displaystyle S = \sum_{r=1}^{d} r\cdot 2^r$$
$$\displaystyle 2S = \sum_{r=1}^{d} r\cdot 2^{r+1} = \sum_{r=2}^{d+1} (r-1)2^{r}$$
$$\displaystyle 2S -S = d\cdot 2^{d+1} - \sum_{r=1}^{d} 2^r = d\cdot 2^{d+1} - 2^{d+1} +2 = (d-1)2^{d+1} + 2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/11464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 9,
"answer_id": 5
} |
Average length of the longest segment This post is related to a previous SE post If a 1 meter rope …. concerning average length of a smallest segment.
A rope of 1m is divided into three pieces by two random points. Find the average length of the largest segment.
My answer is 11/18. Here is how I do it:
Here we hav... | Neat as it is, I don't think Rahul's answer can be correct. If we have $3x+2y+z=1$ and $3x+2y+z=9n$, then $n=1/9$, which means $x \leq 2/9$, which can't be right, as one solution is all pieces being of length 1/3 (however unlikely this exact solution may be, $x$ can take values in $(2/9,1/3]$).
Stefan's answer is wrong... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/14190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 3
} |
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$ Show that if $a$, $b$ are positive integers, then we have:
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$.
| Another Dubuquesque attempt; for legibility, write $d=\gcd(a,b)$:
\begin{align*}
\gcd\Bigl(d(a+b), ab\Bigr) &= \gcd\Bigl(d(a+b), ab, ab\Bigr)\\
&=\gcd\Bigl(d(a+b),\ ab-a(a+b),\ ab-b(a+b)\Bigr)\\
&=\gcd\Bigl(d(a+b),\ a^2,\ b^2\Bigr)\\
&=\gcd\Bigl(d(a+b),\ \gcd(a^2,b^2)\Bigr)\\
&=\gcd\Bigl(d(a+b),\ \gcd(a,b)^2\Bigr)\\
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
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How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$? I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
| Just for completeness, here is induction (just for divisibility by 7) :
Claim : $n^7 - n$ is divisible by 7
Base Case: True for n = 1,2
Induction Step:
Assume true for n = k. To prove true for n = k + 1.
Now, $$(k+1)^7 - (k+1) = k^7 + 7k^6 + 21k^5 + 35k^4 + 35k^3 + 21k^2 + 7k + 1 - k - 1 \\= (k^7 - k) + 7(k^6 + 3k^5 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 8,
"answer_id": 4
} |
How to prove a formula for the sum of powers of $2$ by induction? How do I prove this by induction?
Prove that for every natural number n, $ 2^0 + 2^1 + ... + 2^n = 2^{n+1}-1$
Here is my attempt.
Base Case: let $ n = 0$ Then, $2^{0+1} - 1 = 1$ Which is true.
Inductive Step to prove is: $ 2^{n+1} = 2^{n+2} - 1$
Our h... | I don't see the answer I like here, so I'm writing my own.
Basic proof:
We wish to prove $2^0 + 2^1 + ... + 2^{n-1} = 2^n - 1$ for all $n$. We can verify by inspection this is true for n=1. Next, assume that $2^0 + 2^1 + ... + 2^{n} = 2^{n+1} - 1$.
$(2^0 + 2^1 + ... + 2^n) + 2^{n+1} = (2^{n+1} - 1) + 2^{n+1} = 2 \cdot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/22599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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"answer_id": 3
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Show that $3^{4n+2} + 1$ is divisible by $10$ I'm am a little bit stuck on this question, any help is appreciated.
Show that for every $n\in\mathbb{N}$, $3^{4n+2} + 1$ is divisible by $10$.
| What is the formation law of the remainders of the division by $10$ of the
powers $3^{n}$?
(For the notation see modular arithmetic.)
$$\left\{
\begin{array}{c}
3\equiv 3\quad \pmod{10} \\
3^{2}\equiv 9\quad \pmod{10} \\
3^{3}\equiv 7\quad \pmod{10} \\
3^{4}\equiv 1\quad \pmod{10}
\end{array}\right. $$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/24262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
} |
Implicitly find the second derivative Given the formula $x^{2}y^{2}-8x=3$, find the second derivative.
I calculated the first derivative as $$-\frac{xy^{2}+4}{x^{2}y}$$
Working from that, I calculated the second derivative starting with
$$\frac{([x^{2}y)\frac{d}{dx}(-xy^{2}+4)]-[(-xy^{2}+4)\frac{d}{dx}(x^{2}y)]}{(x^{2}... | It's actually a bit simpler to work directly with the original (since then you don't have to worry about the quotient rule). Simply take derivatives twice, and then solve for $y''$ in terms of $x$, $y$, and $y'$; only then plug in $y'$.
Start with
$$x^2y^2 - 8x=3.$$
Taking derivatives once, we get
\begin{align*}
\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/25840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $p - a \equiv -a \pmod{p}$ then what is $\frac{p-1}{2} \equiv ? \pmod{p}$? If $p - a \equiv -a \pmod{p}$ then what is $\frac{p-1}{2} \equiv ? \pmod{p}$? Where $p$ is an odd prime.
I read in the book, they claimed:
$$p - 1 \equiv -1 \pmod{p}$$
$$p - 2 \equiv -2 \pmod{p}$$
$$p - 3 \equiv -3 \pmod{p}$$
$$ ... $$
$$\fra... | In the 5th edition, they have got the signs right, at least thats what the scanned version of the ebook says.
EDIT
We have $p-k \equiv -k \pmod{p}$ and hence $$ \prod_{k=1}^{\frac{p-1}{2}} (p-k) \equiv \prod_{k=1}^{\frac{p-1}{2}} (-k) \pmod{p}$$
Note that $$(p-1)! = \left(\prod_{k=1}^{\frac{p-1}{2}} k \right) \left(\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/25913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Find the value of 'x' in a product of exponentiated logarithms Find the value of '$x$' if, $$\large \left(\frac{1}{2^{\log_x 4}} \right) \cdot \left( \frac{1}{2^{\log_x 16}} \right) \cdot \left(\frac{1}{2^{\log_x 256}} \right) \cdots = 2 $$
I tried to make it simple by resulting series is not converging, the suggested ... | $$
\begin{align}
\left(\frac{1}{2^{\log_x 4}}\right)\left(\frac{1}{2^{\log_x 16}}\right)\left(\frac{1}{2^{\log_x16}}\right)\cdots&=2\\
2^{-\log_x 4}\times2^{-\log_x 16}\times2^{-\log_x 256}\ldots&=2\\
2^{-\left(\log_x 4+\log_x 16+\log_x 256+\cdots\right)}&=2\\
\log_x(4\times16\times256\times\cdots)&=-1\\
4\times ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/27928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Generating functions in combinatorics I am not very familiar with how generating functions are used but for something I was needing I ran into the following constructions,
*
*Let $c_{nk}$ be the number of solutions in $\{1,2,3,4...,\}$ for the equation $x_1+x_2+x_3+...+x_k = n$. Then one can easily show that $c_{nk... | In both cases, the key is to recognize the power series identity (which should be familiar if you know about geometric series)
$$ 1+y+y^2+y^3+ \ldots = \sum_{i=0}^\infty y^i = \frac{1}{1-y}. $$
In your formula for $c_k(x)$, the last exponent should certainly be negative, so we have $c_k(x) = x^k(1-x)^{-k}$. Then appl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proof that there are infinitely many primes of the form $4m+3$ I am reading a proof of there are infinitely many primes of the form $4m+3$, but have trouble understanding it. The proof goes like this:
Assume there are finitely many primes, and take $p_k$ to be the largest prime of the form $4m+3$.
Let $N_k = 2^2 \cdo... | For the first question, if $N_k=2^2\cdot 3\cdot 5\cdots p_k-1$, then
$$
N_k-3=2^2\cdot 3\cdot 5\cdots p_k-1-3=4(3\cdot 5\cdots p_k-1)
$$
which implies $4|N_k-3$, that is, $N_k\equiv 3\pmod{4}$.
For the second question, suppose instead that $N_k$ has no prime factors of form $4m+3$. Then all its prime factors must be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
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How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that ther... | Note that $\int \{1 + 2x + 3x^2 + \cdots\} \, dx = x + x^2 + x^3 + \cdots + \text{const}$, i.e., a geometric series, which converges to $x/(1 - x)$ if $|x| < 1$. Therefore,
$$\frac{d}{dx} \left(\frac{x}{1 - x}\right) = \frac{(1 - x)(1) - x(-1)}{(1 - x)^2} = \frac{1}{(1 - x)^2},$$
that is,
$$1 + 2x + 3x^2 + \cdots = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\lfloor \sqrt{p} \rfloor + \lfloor \sqrt{2p} \rfloor +...+ \lfloor \sqrt{\frac{p-1}{4}p} \rfloor = \frac{p^2 - 1}{12}$ Problem
Prove that $\lfloor \sqrt{p} \rfloor + \lfloor \sqrt{2p} \rfloor +...+ \lfloor \sqrt{\frac{p-1}{4}p} \rfloor = \dfrac{p^2 - 1}{12}$ where $p$ prime such that $p \equiv 1 \pmod{4}$.
... | The sum $S(p)$ counts the lattice points with positive coordinates under $y=\sqrt{px}$ from $x=1$ to $x=\frac{p-1}{4}$. Instead of counting the points below the parabola, we can count the lattice points on the parabola and above the parabola, and subtract these from the total number of lattice points in a box. Stop her... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 1,
"answer_id": 0
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Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ How can one show that the number
$2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$
is a root of the equation
$\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
| This solution is basically the stupidest possible way to solve the problem. I'm just posting it in case the direct calculation here sheds any light on the problem, it probably doesn't though. Especially frustrating is this does not seem to show any connection between the discriminant $19^2$ and the cyclotomic field inv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/31352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Question regarding the product of quadratic residue modulo $p$, $p$ is prime Problem
Prove that the product of the quadratic residue modulo $p$ is congruent to $1$ modulo p if $p \equiv -1 \pmod{4}$ and is congruent to $-1$ modulo $p$ if $p \equiv 1 \pmod{4}$.
First of all, my question is what do they mean by produc... | They mean product as in multiplication, e.g. the product of the numbers $a$, $b$, and $c$ is $abc$.
Hint: If $x$ is a quadratic residue modulo $p$, then so is $x^{-1}$ (recall that $x^{-1}$ is the $a$ such that $ax=1\bmod p$). Pair up each quadratic residue with its inverse in the product, and cancel each pair; what's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/33333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area of a circle externally tangent to three mutually tangent circles Given three identical circles, with three points of intersection. The line between two of these intersecting points is $3$ feet. They are inside a $4$th circle. All circles are tangent to each other.
What is the area of the $4$th circle? I don't und... | Given that the radii of the inner circles are $3$ feet, Descartes' Theorem (aka Soddy's Theorem) says that
$$
\left(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}\right)^2=2\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r_4^2}\right)
$$
Plugging in $r_1=r_2=r_3=3$, we get
$$
\left(1+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/36353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What kind of series is this, and how do I sum it? $\displaystyle\sum{\frac1{a_n}}$, e.g.
$$\frac 1 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \dots + \frac 1 n$$ or
$$\frac 1 2 + \frac 1 4 + \frac 1 6 + \frac 1 8 + \dots + \frac 1 {2n}$$
| A nice approximation for $n \to \infty$ can be derived from
$$\eqalign{
& \log \left( {1 + x} \right) = \sum\limits_{k = 0}^\infty {{{\left( { - 1} \right)}^k}\frac{{{x^{k + 1}}}}{{k + 1}}} = x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{3} - + \cdots \cr
& \log \left( {1 + \frac{1}{n}} \right) = \sum\limits_{n = 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Computing the integral of $\log(\sin x)$ How to compute the following integral?
$$\int\log(\sin x)\,dx$$
Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of in... | There was a duplicate posted a while ago. Since I think my answer might be of some interest, here it goes:
By substituting $\sin{x}=t$, we can write it as:
\begin{align*}
\int_{0}^{\pi/2} \, \log\sin{x}\, dx &= \int_{0}^{1} \, \frac{\log{t}}{\sqrt{1-t^2}}\, dt \tag{1}
\end{align*}
Now, consider:
\begin{align*}
I(a)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 11,
"answer_id": 2
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Solve recursion $a_{n}=ba_{n-1}+cd^{n-1}$ Let $b,c,d\in\mathbb{R}$ be constants with $b\neq d$. Let
$$\begin{eqnarray}
a_{n} &=& ba_{n-1}+cd^{n-1}
\end{eqnarray}$$
be a sequence for $n \geq 1$ with $a_{0}=0$. I want to find a closed formula for this recursion. (I only know the german term geschlossene Formel and tran... | Argh... use Wilf's techniques from "generatingfunctionology". Start defining:
$$
A(z) = \sum_{n \ge 0} a_n z^n
$$
Also write:
$$
a_{n + 1} = b a_n + c d^n
$$
Multiply by $z^n$, add over $n \ge 0$:
$$
\frac{A(z) - a_0}{z} = b A(z) + c \frac{1}{1 - d z}
$$
Solve for $A(z)$, express as partial fractions. The resulting ter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/38543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
} |
Polynomial equations with finite field arithmetic there are given 3 equations (they are connected with cyclic codes):
$$s(x)=v(x)+q(x)g(x)$$
$$g(x)h(x)=x^7+1$$
$$s(x)=v(x)h(x)\bmod(x^7+1)$$
I have following data (for $GF(8)$ with generator polynomial $p(x)=x^3+x+1$):
$$g(x)=x^4+\alpha^3x^3+x^2+\alpha x+\alpha^3$$
$$q(x... | Your third equation should be $\rm\ s(x)\ h(x)\equiv v(x)\ h(x)\ \ (mod\ x^7 + 1)\:,\:$ which agrees with the calculations. It arises by multiplying the first equation by $\rm\:h(x)\:,\:$ then using the second equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Understanding a simplification in a theorem I'm trying to understand a theorem in a paper on page 14/24. We are given that
$$Z = (nq-1) \log \left(\frac{M+nq-1}{nq-1} \right) + M \log \left(\frac{M+nq-1}{M} \right) + \frac{1}{2} \log \left( \frac{M+nq-1}{(nq-1)M} \right) + O(1) .$$
Since $e^a > \left(1+\frac{a}{b} \ri... | OK, thanks to the paper, I figured it out. What they do is rewrite
$$Z = (nq-1) \log \dfrac{M+nq-1}{nq-1} + M \log \dfrac{M+nq-1}{M} + \dfrac{1}{2} \log \dfrac{M+nq-1}{(nq-1)M} + O(1)$$
with their formula for the second term
$$M\log\left(1 + \dfrac{nq-1}{M}\right) < (nq-1)\log e$$
which gives
$$Z < (nq-1) \log \dfrac{M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all prime $p$ such that $x^2=-1$ has a solution in $\mathbb{Z}/p\mathbb{Z}$ I have found by a numerical experiment that first such primes are:
$2,5,13,17,29,37,41$. But I cannot work out the general formula for it.
Please share any your ideas on the subject.
| The following argument depends on knowing (or separately proving) Wilson's Theorem.
Theorem Let $p$ be prime. Then $(p-1)! \equiv -1 \pmod{p}$.
Now we use Wilson's Theorem to prove the result. For the sake of concreteness, I will use $p=17$, but the same idea exactly works for all primes congruent to $1$ modulo $4$. A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Find $\sin \theta$ and $\cos \theta$ given $\tan 2\theta$ Can you guys help with verifying my work for this problem. My answers don't match the given answers.
Given $\tan 2\theta = -\dfrac{-24}{7}$, where $\theta$ is an acute angle, find $\sin \theta$ and $\cos \theta$
I used the identity, $\tan 2\theta = \dfrac{2\t... | At Chandru's request:
*
*The quadratic $12z^2-7z-12$ factors as $(3z-4)(4z+3)$ so we should get $\tan\,\theta=4/3$ and $\tan\,\theta=-3/4$.
*"Acute angle" means "angle between 0 and $\pi/2$" means 1st quadrant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/49569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Trigonometric equality: $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$ Can you guys give me a hint on how to proceed with proving this trigonometric equality? I have a feeling I need to use the half angle identity for $\tan \frac{\theta}{2}$. The stuff I have tried so far(multiplying numerator and... | Now that OP has understood how to prove this, here is a geometric proof for certain angles, just for fun :-)
Consider the figure:
$\displaystyle \triangle ABC$ is a right angled triangle with the right angle being at $\displaystyle C$.
$\displaystyle \angle{CAB} = A$ and $\displaystyle AB = 1$ and thus $\displaystyle ... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 4
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Is this Batman equation for real? HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{... | Looking at the equation, it looks like it contains terms of the form
$$ \sqrt{\frac{| |x| - 1 |}{|x| - 1}} $$
which evaluates to
$$\begin{cases} 1 & |x| > 1\\ i & |x| < 1\end{cases} $$
Since any non-zero real number $y$ cannot be equal to a purely imaginary non-zero number, the presence of that term is a way of writin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "466",
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Can $A^2+B^2+C^2-2AB-2AC-2BC$ be a perfect square? I have come upon the trivariate polynomial $A^2+B^2+C^2-2AB-2AC-2BC$ and want to factor it. Because of the symmetry, I am wondering if it can be a perfect square or if there is some other nice factorization.
| A more symmetric factorization would be:
$$A^2 + B^2 + C^2 - 2AB -2AC - 2BC =$$
$$(A + B + C)^2 - 4AB - 4AC - 4BC =$$
$$(A + B + C - 2 \sqrt{AB + AC + BC}) (A + B + C + 2 \sqrt{AB + AC + BC})$$
if
$$AB + AC + BC \ge 0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many real roots are there to $2^x=x^2$? How many real roots are there to
$2^x=x^2$?
| An obvious solution is $x=2$.
If $2^x = x^2$, then $x\neq 1$ and $x\neq 0$. I'll treat the positive and negative cases separately.
If $x\gt 0$, then we get $x\ln(2) = 2\ln (a)$, or $\frac{x}{\ln x} = \frac{2}{\ln 2}$.
The derivative of $g(x) = \frac{x}{\ln x}$ is $\frac{\ln x - 1}{(\ln x)^2}$.
On $(1,\infty)$, the deri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is numerator re-defining necessary for proper solution of some partial fraction integrations? In my homework, one problem was the following formula. Using standard partial fraction techniques where you'll see my work, I came up with an almost correct answer in the fact the book solution had a three term integral r... | Normally one uses the division algorithm to decompose into integral and proper fractional parts
$$\rm \dfrac{2x^3-4x^2-15x+5}{(x+2)\:(x-4)}\ =\ 2\:x + \dfrac{x+5}{(x+2)\:(x-4)} $$
Then one performs the partial fraction decomposition only on the second fractional part. But you seem to desire to skip the initial division... | {
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Whether this matrix is positive definite Let $A$ be a nonsingular real square matrix. Is it true that the matrix
$$\frac{1}{2}(A+A')-2(A^{-1}+(A^{-1})')^{-1}$$
is positive semidefinite?
Here, $A'$ denotes the transpose of $A$.
Edited
Let $A,B$ be positive definite matrices of the same size, is it true that $\frac{1}... | Added: This answer the updated question.
Your updated conjecture is also false. Consider:
$$
A = \left(
\begin{array}{cc}
10 & -5 \\
9 & 3 \\
\end{array}
\right) \qquad\qquad\qquad
B = \left(
\begin{array}{cc}
1 & -6 \\
8 & 8 \\
\end{array}
\right)
$$
Then
$$
\frac{1}{2}(A.B + B.A) - 2 ( (A.B)^{-1} + (B.A)^{... | {
"language": "en",
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How to find the minimum value of $\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}$? Let $x,y,z\in [1,4]$ such that $x \geq y$ and $x \geq z$.
Find the minimum value of this expression:
$$
P=\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}
$$
| As has been mentioned in comments, we want to find the minimum value of
$$
P=\frac{1}{2+3u}+\frac{u}{u+v}+\frac{v}{v+1}
$$
for $u,v\in[\frac{1}{4},1]$. Take partials of $P$ with respect to $u$ and $v$:
$$
\begin{align}
\frac{\partial P}{\partial u}&=-\frac{3}{(2+3u)^2}+\frac{v}{(u+v)^2}\\
\frac{\partial P}{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/59814",
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Restricted Integer Compositions Let $c_{k}(N;[a,b])$ denote the number of compositions of $N$ into $k$ parts, where each part is restricted to the interval $[a,b]$, i.e., $N = \sum_{i = 1}^{k} s_{i}$ with $a \leq s_{i} \leq b$. The generating function of $c_{k}(N;[a,b])$ is
\begin{align}
G(c_{k}(N; [a,b]);t) = t^{ka} ... | The $\alpha_i$ really contribute nothing and can simply be subtracted from $N$.
For the rest, if $\gamma_i$ divides $\beta_i$, you are looking for compositions of $N$ (minus the sum of the alphas) that start with $k_1$ terms at most $\beta_1$ and divisible by $\gamma_1$ then $k_2$ terms with the analogous properties an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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evaluating $ \int_0^{\sqrt3} \arcsin(\frac{2t}{1+t^2}) \,dt$ $$\begin{align*}
\int \arcsin\left(\frac{2t}{1+t^2}\right)\,dt&=t\arcsin\left(\frac{2t}{1+t^2}\right)+\int\frac{2t}{1+t^2}\,dt\\
&=t\arcsin\left(\frac{2t}{1+t^2}\right) + \ln(1+t^2)+C
\end{align*}$$
So
$$ \int\nolimits_0^{\sqrt3} \arcsin\left(\frac{2t}{1+t^... | I think you made a simplification error. We have
$$\begin{align*}
\frac{d}{dt}\arcsin\left(\frac{2t}{1+t^2}\right) &= \frac{1}{\sqrt{1 - \frac{4t^2}{(1+t^2)^2}}}\left(\frac{2t}{1+t^2}\right)'\\
&= \frac{(1+t^2)}{\sqrt{(1+t^2)^2-4t^2}}\left(\frac{2(1+t^2)-4t^2}{(1+t^2)^2}\right)\\
&= \frac{(1+t^2)}{\sqrt{(1-t^2)^2}}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How would I solve $\frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376$ for some $n$ without brute forcing it? Given this equation:
$$
\frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376
$$
How would I find $n$?
I already know the answer to this, all thanks toWolfram|Al... | Using @pharmine's factorization, we have
$$\begin{eqnarray*}
n(n-1)(n-2)\cdots(n-10) &=& 17\cdot 13 \cdot 7 \cdot 2^3 \cdot 11! \\
&=& 17\cdot 13 \cdot 7 \cdot 2^3 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2
\end{eqnarray*}$$
Now, $17$ divides the LHS, so LHS consists of eleven conse... | {
"language": "en",
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"source": "stackexchange",
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Distributive Property on Fractions: Swapping Denominators I'm learning Algebra and am curious about some methodological fundamentals here. One, in particular is why the following equation:
6(2x + 1 / 3) = 6(x + 4 / 2)
results in:
2(2x + 1) = 3(x + 4)
It's obvious that the distributive property swaps the numerators of... | May be the following steps help you see how you get the result from the given expression - Swapping is fine as long as you understand the meaning of it so that you don't make mistakes.
Given:
$6 \left ( \frac{2x+1}{3} \right )= 6 \left ( \frac{x+4}{2} \right )$
a-multiply both sides by 1/6 to get:
$ \left ( \frac{2x+1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do definitions of distinct conic sections produce a single equation? I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and $(c,0)$ is constant, $2a$ — and an ellipse — as the ... | This is the difference between real numbers, some of which are not squares of other real numbers, and complex numbers, all of which are squares of other complex numbers. Is the $a^2 - c^2$ in the denominator a negative number (i.e. one that is not a square) or a positive number (one that is a square) (the case where i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Problem about sum of arithemtic progression and geometric progression Question: An arithmetic sequence has a common difference of $1$ and a geometric sequence has a common ratio of $3$. A new sequence is formed by adding corresponding terms of these two progressions. It is given that the second and fourth term of the s... | (i)
the first sequence is: $u_n=u_0+n$
the second sequence is $v_n=v_0 \cdot 3^n$
so the new sequence is : $w_n=u_0+n+v_0 \cdot 3^n$
since $w_2=12$ and $w_4=86$
then $u_0+2+v_0\cdot9=12$ and $u_0+4+v_0\cdot81=86$
then $u_0=1$ and $v_0=1$
so $w_n=1+n+3^n$
(ii)
$\sum\limits_{k=0}^{n-1}w_k = \sum\limits_{k=0}^{n-1}1+\su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of dividing fractions $\frac{a/b}{c/d}=\frac{ad}{bc}$ For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$
| Write $\frac{a}{b} \div \frac{c}{d}$ as
$$
\frac{\ \frac{a}{b}\ }{\frac{c}{d}}.
$$
Suppose you wanted to clear the denominator of this compound fraction. You could try multiplication by $\frac{d}{c}$, but you'll have to multiply the top and the bottom of the fraction to avoid changing it. So, you end up with
$$
\fra... | {
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How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$? How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$?
Should I use some geometrical approach or apagoge?
| If $\cos \frac{2 \pi}{7}$ is rational, then $i\sin \frac{2 \pi}{7}=\sqrt{\cos^2 \frac{2 \pi}{7}-1}$ is a quadratic irrational, and hence so is $\cos \frac{2 \pi}{7}+i\sin \frac{2 \pi}{7}$. But $\cos \frac{2 \pi}{7}+i\sin \frac{2 \pi}{7}$ is a primitive $7$th root of $1$, and so has minimal polynomial $x^6+x^5+x^4+x^3+x... | {
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$1\cdot 3 + 2\cdot 4 + 3\cdot 5 + \cdots + n(n+2) = n(n+1)(2n+7)/6$ by mathematical induction I am doing mathematical induction. I am stuck with the question below. The left hand side is not getting equal to the right hand side.
Please guide me how to do it further.
$1\cdot 3 + 2\cdot 4 + 3\cdot 5 + \cdots + n(n+2) = ... | You used the wrong formula in your induction hypothesis. You are assuming that
$$1\cdot 3 + 2\cdot 4 + \cdots + k(k+2) = \frac{1}{6}k(k+1)(2k+7)$$
but in your inductive argument, you wrote
$$1\cdot 3 + 2\cdot 4 + \cdots + k(k+2) + (k+1)(k+3) = \frac{1}{6}(k+1)(k+2)(2k+9) + (k+1)(k+3).$$
That is, you substituted $1\cdot... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $x$ if $(7+4\sqrt 3)^{x^2-8}+(7-4\sqrt 3)^{x^2-8}=14$
Known that:
$$(7+4\sqrt 3)^{x^2-8}+(7-4\sqrt 3)^{x^2-8}=14$$
What is the value of $x$
| Let $a=(7+4\sqrt{3})^{x^2-8}$ then $\dfrac{1}{a}=(7-4\sqrt{3})^{x^2-8}$
So the given equation will be $a+\dfrac{1}{a}=14\implies a^2-14a+1=0$
Solving quadratic for $a=7+4\sqrt{3}$ or $a=7-4\sqrt{3}\implies x^2-8=1$ or $-1$
therefore possible values of $x=3,-3,\sqrt{7},-\sqrt{7}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the formula for $n^{th}$ derivative of $ \sin^{-1} x, \quad \tan^{-1} x,\quad \sec x \quad \text{and}\quad \tan x$? Are there formulae for the nth derivatives of the following functions?
$1)\quad$ $sin^{-1} x$
$2)\quad$ $tan^{-1} x$
$3)\quad$ $sec x$
$4)\quad$ $tan x$
Thanks.
| A related problem. See Chapter 6 in this book for formulas for the nth derivative, $n$ is a non-negative integer, of $\tan(x)$ and $\sec(x)$ in terms of the $\psi$ function
\begin{equation}
{\tan}^{(n)} (z) = \frac{1}{{\pi}^{n+1}} \left({\psi}^{(n)} \left(
\frac{1}{2}+\frac{z}{\pi}\right) + (-1)^{n+1} {\psi}^{(n)} \le... | {
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If $x^2+y^2=z^2$, why can't $x$ and $y$ both be odd? What does the following mean: If $x^2 + y^2 = z^2$ some integers $z$, then $x$ and $y$ can't be both odd (otherwise, the sum of their squares would be $2$ modulo $4$, which can't be a square). So, one of them must be even?
I see that if $x$ and $y$ are both odd, th... | If $n = 2k$, then $n^2 = 4k^2$ is a multiple of $4$.
Likewise, if $n = 2k+1$, then $n^2 = 4(k^2 + k) + 1$.
Therefore, a square is congruent to either $0$ or $1 \pmod{4}$. In other words, a square is never of the form $4k + 2$, for some $k$.
More specifically, since you've seen that $x^2 + y^2 \equiv 2 \pmod{4}$ wh... | {
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"source": "stackexchange",
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Pythagorean quadruples Another Project Euler problem has me checking the internet again. Among other conditions, four of my variables satisfy:
$$a^2+b^2+c^2=d^2 .$$
According to Wikipedia, this is known as a Pythagorean Quadruple. It goes on to say all quadruples can be generated from an odd value of $a$ and an even ... | I think $c$ and $d$ should have been
$$
\begin{split}
c &= \frac{a^2+b^2-p^2}{2 p}\qquad\qquad
d &= \frac{a^2+b^2+p^2}{2 p}
\end{split}
$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding $\lim\limits_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}$ $$\lim_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}.$$
With a first look this must give $1$ as a result but have a problem to explain it.
How can I do it?
Edit
I noticed that it is $\frac{\infty}{\infty}$.
$$\lim_{n \to \infty}{n^{n}\frac{(\... | Let $f(n) = (1^1 + 2^2 + 3^3 + \cdots + n^n)/n^n$. You want to show $\lim_{n \to \infty} f(n) = 1$.
It's obvious that $f(n) > 1$ for all $n$.
For an upper bound,
$$ f(n) \le {1^{n-2} + 2^{n-2} + \cdots + (n-2)^{n-2} \over n^n} + {n^{n-1} \over n^n} + {n^n \over n^n} = {1^{n-2} + 2^{n-2} + \cdots + (n-2)^{n-2} \over n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
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Find all integers $m$ such that $\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} $ How would you determine all integers $m$ such that the following is true?
$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$
Note that $\lfloor \cdot \rfloor$ means the greatest integer fun... | Consider the reciprocal equation
$$ m = \dfrac{\lfloor 2x \rfloor \cdot \lfloor 5x \rfloor}
{\lfloor 2x \rfloor + \lfloor 5x \rfloor}$$
Letting $x=n+\delta$ where $0 \le \delta < 1$, we get
$$ m = \dfrac{(2n + \lfloor 2\delta \rfloor)(5n + \lfloor 5\delta \rfloor)}
{7n + \lfloor 2\delta \rflo... | {
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How can I prove the inequality $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z}$? For $x > 0$, $y > 0$, $z > 0$, prove:
$$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z} .$$
I can see that this is true, I also checked it with a few numbers. But I guess that is not enough to prove it. So how ... | You can also brute-force it along with some careful re-grouping of terms:
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\geq \frac{9}{x+y+z}$
$\Leftrightarrow \frac{xy+xz+yz}{xyz}\geq \frac{9}{x+y+z}$
$\Leftrightarrow x^2 y + xyz + x^2 z + x y^2 + xyz + y^2 z + xyz + xz^2 + y z^2 \geq 9xyz$
$\Leftrightarrow x^2y-2xyz + yz^2 + x^... | {
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The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0? Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
| The following requires no tools other than induction. In particular, machinery from the calculus is not used.
Let $f(n)$ be the product up to the term $1-\dfrac{1}{2^n}$.
We show by induction that
$$f(n)\ge\frac{1}{4} +\frac{1}{2^{n+1}}.$$
The result is true at $n=1$. For the induction step, note that
$$f(m+1)=f(m... | {
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"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Any trick on finding the inverse of this matrix? Supposing I have a matrix, $\pmatrix{0&0&\lambda\\0&\lambda&-1\\ \lambda&-1&0}$.
Without question you can work out the inverse if this matrix. But since it is highly structured, I suppose there should be some quick way to find out the inverse of it?
Can anyone show me a ... | You can partition the matrix into
$$ \begin{bmatrix} 0 & v^\top \\ v & A \end{bmatrix} $$
where $A$ is a symmetric $2\times2$ matrix and $v$ a $2\times1$ vector, yielding the solution of
$$ \begin{bmatrix} 0 & v^\top \\ v & A \end{bmatrix}^{-1} = \begin{bmatrix} -(v^\top A^{-1} v)^{-1} & T^\top \\ T & A^{-1}-T v^\top A... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Finding the slope of the tangent line to $\frac{8}{\sqrt{4+3x}}$ at $(4,2)$ In order to find the slope of the tangent line at the point $(4,2)$ belong to the function $\frac{8}{\sqrt{4+3x}}$, I choose the derivative at a given point formula.
$\begin{align*}
\lim_{x \to 4} \frac{f(x)-f(4)}{x-4} &=
\lim_{x \mapsto 4}... | All these answers are simple fact (except using L' Hospital, which is the best approach though) that you have to multiply by conjugate surds, which is greatly illustrated in @Michael Hardy's answer. So to sum up, you must make a factor $(x-4)$ in numerator, in order to cancel the same at denominator. But, if that facto... | {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Asymptotic formula for $\sum_{n \le x} \frac{\varphi(n)}{n^2}$ Here is yet another problem I can't seem to do by myself... I am supposed to prove that
$$\sum_{n \le x} \frac{\varphi(n)}{n^2}=\frac{\log x}{\zeta(2)}+\frac{\gamma}{\zeta(2)}-A+O \left(\frac{\log x}{x} \right),$$
where $\gamma$ is the Euler-Mascheroni cons... | Consider Dirichlet series, related to the problem at hand:
$$
g(s) = \sum_{n=1}^\infty \frac{\varphi(n)}{n^{2+s}} = \frac{\zeta(s+1)}{\zeta(s+2)}
$$
We can now recover behavior of $A(x) = \sum_{n \le x} \frac{\varphi(n)}{n^s}$ by employing Perron's formula, using $c > 0$:
$$
A(x) = \frac{1}{2 \pi i} \int_{c - i... | {
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"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
} |
System of first-order ordinary differential equations solve for $x,y,z$:
$$\frac{dx}{x^{2}+a^{2}}=\frac{dy}{xy-az}=\frac{dz}{xz+ay}$$
please give a hint. I am not able to formulate the steps required to proceed solving this one.
| We have
$$
\begin{align}
\frac{dy}{xy-az} & = \frac{dz}{xz+ay}\\
\frac{dy/y}{x-a (z/y)} & = \frac{dz/z}{x+a (y/z)}
\end{align}
$$
This gives a motivation to let $z = ky$ where $k$ is a constant.
$$
\begin{align}
\frac{dy/y}{x-a k} & = \frac{dy/y}{x+a/k}
\end{align}
$$
This gives us that $k = \pm i$. Let $k=i$... | {
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"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares Can we find pairs $(x,y)$ of positive integers such that $x^2+3y$ and $y^2+3x$ are simultaneously perfect squares? Thanks a lot in advance. My progress is minimal.
| Not a complete solution but an approach that seems like it will work.
Assume $y \gt x$
Then we have that
$(y+2)^2 \gt y^2 + 3y \gt y^2+3x \gt y^2$
If $y^2 + 3x$ was a perfect square, then we have that $y^2 + 3x = (y+1)^2$.
This gives us $3x = 2y+1$.
Substitute in the other expression, and form similar inequalities. Th... | {
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"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
How to find the roots of $f(x)= \ln( \frac{x+1 }{x-2})$? I can't solve this equation:
$$\ln\left(\frac{x+1}{x-2}\right) = 0.$$
I do:
$$\begin{align*}
\ln \left( \frac{x+1}{x-2} \right)&=0\\
\frac{x+1}{x-2} &= 1 \\
x+1&=x-2 \\
x+1-x+2&=0 \\
x-x+3&=0 \\
3&=0
\end{align*}$$
Then $x$ is?
| What you've shown is that $\frac{x+1}{x-2}$ is never equal to 1. Since 1 is the only value where natural log equals zero, the equation $\log \frac{x+1}{x-2} = 0 $ has no solutions.
| {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to prove $(1+1/x)^x$ is increasing when $x>0$? Let $F(x)=(1+\frac{1}{x})^x$.
How do we prove $F(x)$ is increasing when $x>0$?
| This proof is from theorem 140 from Hardy's Inequalities.
Let $f(x) = \ln\left[\left(1 + \frac{1}{x}\right)^x\right] = x(\ln(x+1) - \ln(x)).$ We refer to the mean value theorem: for each differentiable $g$,
$$
g(x + h) - g(x) = hg'(x + \theta h)
$$
for $\theta \in (0,1)$.
Applying the MVT to $g(x) = \ln(x)$, we get $\l... | {
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"source": "stackexchange",
"question_score": "23",
"answer_count": 8,
"answer_id": 6
} |
Proving $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$ Could anyone help me prove that $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$?
As $6=2*3=(1+\sqrt{-5})(1-\sqrt{-5})$ so $\mathbb{Z}[\sqrt{-5}]$ is not a UFD. Therefore is not a PID or euclidean domain... | The standard method is:
Define a function $N\colon \mathbb{Z}[\sqrt{-5}]\to\mathbb{Z}$ by $N(a+b\sqrt{-5}) = (a+b\sqrt{-5})(a-b\sqrt{-5}) = a^2+5b^2$.
*
*Prove that $N(\alpha\beta) = N(\alpha)N(\beta)$ for all $\alpha,\beta\in\mathbb{Z}[\sqrt{-5}]$.
*Conclude that if $\alpha|\beta$ in $\mathbb{Z}[\sqrt{-5}]$, then ... | {
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"question_score": "28",
"answer_count": 3,
"answer_id": 1
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Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$? Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
| Let $\mathcal{I}(a)$ denote the integral. Then
$$ \begin{eqnarray}
\mathcal{I}(a) &=& \int_0^1 \frac{\mathrm{d} x}{(1+x^2)(1+x^a)} + \int_1^\infty \frac{\mathrm{d} y}{(1+y^2)(1+y^a)} \\
&\stackrel{y=1/x}{=}&
\int_0^1 \frac{\mathrm{d} x}{(1+x^2)(1+x^a)} +
\int_0^1 \frac{x^a \mathrm{d} x}{(1+x^2)(1+x^... | {
"language": "en",
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"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 0
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How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$ We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact,
$$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$$
How do we prove that, ... | I'll copy here an answer, which I posted before at AoPS. However the solution from wikipedia article (see the link in lhf's comment to J.M.'s answer) seems to be much more elegant. Basically these matrix proofs often can be rewritten to proofs using generating functions.
We denote by $F_i$ the i-th Fibonacci number.
U... | {
"language": "en",
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"source": "stackexchange",
"question_score": "29",
"answer_count": 3,
"answer_id": 0
} |
$\int \cos^{-1} x \; dx$; trying to salvage an unsuccessful attempt $$
\begin{align}
\int \cos^{-1} x \; dx &= \int \cos^{-1} x \times 1 \; dx
\end{align}
$$
Then, setting
$$\begin{array}{l l}
u=\cos^{-1} x & v=x \\
u' = -\frac{1}{\sqrt{1-x^2}} & v'=1\\
\end{array}$$
Then by the IBP technique, we have:
$$\begin... | The integral
$$
\int \frac{1}{\sqrt{1-x^2}} x\;dx
$$
is BEGGING for a simple substitution. Don't integrate by parts here. Instead, do this:
$$
\begin{align}
u & = 1 - x^2 \\ \\ du & = -2x\;dx \\ \\ \frac{-du}{2} & = x\;dx
\end{align}
$$
You get
$$
-\int \frac{1}{2\sqrt{u}} \;du = \sqrt{u}+C = \sqrt{1-x^2}+ C... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Showing two matrices are similar I have to show that each of the following matrices
$$ \frac{1}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\
1&0&1\\
0&1&0
\end{pmatrix}\quad ,
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0&-i&0\\
i&0&-i\\
0&i&0
\end{pmatrix}
,
\begin{pmatrix}
1&0&0\\
0&0&0\\
0&0&-1
\end{pmatrix}$$
are ... | Two matrices are similar if their traces are equal.
| {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is this a proper use of induction? ($(n^2+5)n$ is divisible by 6) Just want to get input on my use of induction in this problem:
Question. Use mathematical induction to prove that $(n^2+5)n$ is divisible by $6$ for all integers $n \geqslant 1$.
Proof by mathematical induction.
(1) show base case ($n=1$) is true:
$$
... | It is definitively not sufficient, since "letting $n=1$" means "you don't know what to do in every other cases of values of $n$". Since saying that $6$ divides $3n^2 + 3n$ is equivalent to saying that $2$ divides $n^2 + n$, you only need to show the latter. Now why should $2$ divide $n^2+n = n(n+1)$, two consecutive in... | {
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"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Drawing $z^4 +16 = 0$ I need to draw $z^4 +16 = 0$ on the complex numbers plane.
By solving $z^4 +16 = 0$ I get:
$z = 2 (-1)^{3/4}$ or $z = -2 (-1)^{3/4}$ or $z = -2 (-1)^{1/4}$ or $z = 2 (-1)^{1/4}$
However, the suggested solution by my teacher is:
Where you can see that the solutions that he found are:
$z = \sqrt{... | The solutions that your teacher found are the $4$ solutions to $z^4=16$. Here we will find these roots algebraically.
As you point out, the solutions should be of the form $z=2\cdot (-1)^{1/4}$, where "$(-1)^{1/4}$" stands for any fourth root of $-1$. Thus, we need to find the solutions to
$$z^4=-1,$$
or equivalently,... | {
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"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Formula for the sum of the squares of numbers We have the well-known formula
$$\frac{n (n + 1) (2 n + 1)}{6} = 1^2 + 2^2 + \cdots + n^2 .$$
If the difference between the closest numbers is smaller, we obtain, for example
$$\frac{n \times (n + 0.1) (2 n + 0.1) }{6 \cdot 0.1} = 0.1^2 + 0.2^2 + \cdots + n^2 .$$
It is ... | Suppose we want to calculate the sum of squares with successive differences $\epsilon$ from $0$ to some fixed $n$ (we require $\frac{n}{\epsilon}\in\mathbb{N}$ for this particular calculation, however for the general formulation of integrals and Riemann sums, this is not required), that is
$$S_\epsilon = \sum_{i=0}^{\f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Normal and Lower triangular matrix implies diagonal matrix A lower triangular complex matrix $A$ satisfies $AA^*=A^*A$.
I would like to show that $A$ is diagonal. I know there exists a unitary matrix $P$ such that $PAP^*$ is diagonal. But I don't know how to show $A$ itself is diagonal.
| We can show it by induction on the dimension. For $n=2$, let $A=\begin{pmatrix}a&0\\\ b&c\end{pmatrix}$ such a matrix. Then
\begin{align*}A^* A-AA^* &=\begin{pmatrix}\bar a&\bar b\\ 0&\bar c\end{pmatrix}\begin{pmatrix}a&0\\ b&c\end{pmatrix}-\begin{pmatrix}a&0\\ b&c\end{pmatrix}\begin{pmatrix}\bar a&\bar b\\ 0&\bar c\e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$? How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
| Writing $2x^2-1$ as $2x^2 + 2 - 3$, the integral simplifies to
$$\begin{align*}
\int_{-\infty}^{\infty}\exp(-x^2)\frac{2x^2-1}{1+x^2}\mathrm dx
&= 2\int_{-\infty}^{\infty}\exp(-x^2)\mathrm dx
-3 \int_{-\infty}^{\infty}\exp(-x^2)\frac{1}{1+x^2}\mathrm dx\\
&= 2\sqrt{\pi}
- \frac{3}{2\pi}\int_{-\infty}^{\infty}\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
} |
Taylor polynomial of $\int_{0}^{x}\sin(t^2)dt$ I just learned about Taylor polynomials, and I am trying to estimate $\int_{0}^{1/2}\sin(x^2)dx$ using the 3rd degree Taylor polynomial of $F(x)=\int_{0}^{x}\sin(t^2)dt$ at $0$. I get the following:
$F'(x)=\sin(x^2)$,
$F''(x)=2x\cos(x^2)$,
$F^{(3)}(x)=2\cos(x^2)-4x^2\sin(x... | We have
$$f(x)=f(0)+xf'(0)+\frac{x^2}2f''(0)+\frac{x^3}{3!}f^{(3)}(0)+\int_0^x\frac{(x-t)^3}{3!}f^{(4)}(t)dt,$$
so
\begin{align*}
\sin x^2&=x^2+\int_0^x\frac{(x-t)^3}{3!}(8\cos(t^2)(1-5t)+16t^4\sin(t^2))dt\\
&=x^2+\frac 43\int_0^x(x-t)^3(\cos(t^2)(1-5t)+2t^4\sin(t^2))dt,
\end{align*}
and for $x\geq 0$
\begin{align*}
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric equality $\cos x + \cos 3x - 1 - \cos 2x = 0$ In my text book I have this equation:
\begin{equation}
\cos x + \cos 3x - 1 - \cos 2x = 0
\end{equation}
I tried to solve it for $x$, but I didn't succeed.
This is what I tried:
\begin{align}
\cos x + \cos 3x - 1 - \cos 2x &= 0 \\
2\cos 2x \cdot \cos x - 1 - \... | To solve an "equals zero" equation involving polynomials or trigonometric functions, it's generally prudent to keep the $0$ on one side and only work with the other side. Here you could use the double and triple angle formulas (which can be deduced from the addition formulas if need be) given by
$$\cos2x=2\cos^2x-1,\qq... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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$(\cos \alpha, \sin \alpha)$ - possible value pairs We introduced the complex numbers as elements of $ \mathrm{Mat}(2\times 2, \mathbb{R})$ with
$$
\mathbb{C} \ni x =
\left(\begin{array}{cc}
a & -b \\
b & a \\
\end{array}\right) =
\frac{1}{\sqrt{a^2+b^2}}
\left(\begin{array}{cc}
\frac{a}{\sqrt{a^2+b^2}} & \frac... | I suppose you could simply plot the point $(a,b)$ in the $x$-$y$ plane and let $\alpha$ be the angle formed by the positive $x$-axis and the ray joining the origin with $(a,b)$ (measured counterclockwise starting from the positive $x$-axis to the ray).
Then the distance from $(a,b)$ to the origin is $\sqrt{a^2+b^2}$ an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $ Prove this inequality, if $x_1^2+x_2^2+x_3^2=1$:
$$ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $$
So far I got to $x_1^4+x_2^4+x_3^4\ge\frac{1}3$ by using QM-AM for $(2x_1^2+x_2^2, 2x_2^2+x_3^2, 2x_3^2+x... | We assume that $x_i\geq 0$ and let $\theta_i\in \left(0,\frac{\pi}2\right)$ sucht that $x_i=\tan\frac{\theta_i}2$. We have $\sin(\theta_i)=\frac{2x_i}{1+x_i^2}$ and since $\sin$ in concave on $\left(0,\frac{\pi}2\right)$, we have
$$\sum_{i=1}^3\frac{x_i}{1+x_i^2}=\frac 32\sum_{i=1}^3\frac 13\sin(\theta_i)\leq \frac 32... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 0
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System $a+b+c=4$, $a^2+b^2+c^2=8$. find all possible values for $c$. $$a+b+c=4$$$$a^2+b^2+c^2=8$$
I'm not sure if my solution is good, since I don't have answers for this problem. Any directions, comments and/or corrections would be appreciated.
It's obvious that $\{a,b,c\}\in[-\sqrt8,\sqrt8]$. Since two irrational num... | We will show that any $c$ in the closed interval $[0,8/3]$ is achievable, and nothing else is. Let $c$ be any real number, and suppose $a$ and $b$ are real numbers such that $(a,b,c)$ satisfies our two equations. Note that in general
$$2(a^2+b^2)-(a+b)^2=(a-b)^2\ge 0. \qquad\qquad(\ast)$$
Put $a^2+b^2=8-c^2$ and $a+b=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Count the number of divisions of a set recursively I'm trying to understand the reasoning behind the answer to the following question from an old exam.
Given a set $S_n = \{ 1, 2, 3,\dots n\}$ find how many divisions, $K_n$, there are of the set using only subsets of 1 or 2 members. For example, $\{ \{1,4\}, 6 , 2, \{... | Let's take a stab at the recurrence... define the exponential generating function $\hat{K}(z) = \sum_{n \ge 0} K_n \frac{z^n}{n!}$ (to compensate for the $n - 1$ factor), and write:
$$
K_{n + 1} = K_n + n K_{n - 1} \qquad K_0 = K_1 = 1
$$
Using properties of exponential generating functions:
$$
\hat{K}'(z) = \hat{K}(z)... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Writing a function $f$ when $x$ and $f(x)$ are known I'm trying to write a function. For each possible input, I know what I want for output. The domain of possible inputs is small:
$$\begin{vmatrix}
x &f(x)\\
0 & 2\\
1 & 0\\
2 & 0\\
3 &0\\
4 &0\\
5 &0\... | You can generalise the problem: suppose you know the value of $f(x)$ for a particular finite set of values of $x$. (Here, you know the value of $f(x)$ when $x=0,1,2,3,4,5,6$.) Then you can find a possible polynomial function $f$ which takes the given values using the following method.
Suppose you know the value of $f(x... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
} |
Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $ Is there any way to show that
$$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \righ... | Set $x=0$ in the Fourier series of $\cos(ax)$:
$$\cos(ax)=\frac{2a\sin(\pi a)}{\pi}\left[\frac1{2a^2}+\sum_{k=1}^\infty\frac{(-1)^k\cos(kx)}{a^2-k^2}\right],\quad a\notin\mathbb{Z}$$
we get
\begin{align}
\frac{\pi}{\sin(\pi a)}&=\frac1a+\sum_{k=1}^\infty\frac{2a(-1)^k}{a^2-k^2}\\
&=\frac1a+\sum_{k=1}^\infty\frac{(-1)^k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 2
} |
Find smallest number when divided by $2,3,4,5,6,7,8,9,10$ leaves $1,2,3,4,5,6,7,8,9$ remainder Find smallest number when divided by $2,3,4,5,6,7,8,9,10$ leaves $1,2,3,4,5,6,7,8,9$ remainder.How to go about solving this problem??
| The smallest such number is 2519. The next number can obtain by the equation 2519x+x-1. Where x can replaced by any whole number
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/111595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
An integral evaluation I tried my luck with Wolfram Alpha, with $p \in \mathbb{R}$
$$\int_{-\infty}^{\infty} \frac{x^p}{1+x^2} dx = \frac{1}{2} \pi ((-1)^p+1) \sec(\frac{\pi p}{2})$$ for $-1<p<1$, and doesn't exist for other $p$.
I wonder how to integrate it myself? Especially given that $(-1)^p$ may be a non-real comp... | Split integration over $\mathbb{R}$ into integration over $\mathbb{R}_{\geqslant 0}$ and $\mathbb{R}_{<0}$ and perform a change of variables $x \mapsto -x$ in the latter one:
$$
\int_{-\infty}^\infty \frac{x^p}{1+x^2} \mathrm{d} x = \left(1 + (-1)^p \right) \int_0^\infty \frac{x^p}{1+x^2} \mathrm{d} x
$$
Now the id... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integrate using Trigonometric Substitutions Evaluate the integral using trigonometric substitutions.
$$\int{ x\over \sqrt{3-2x-x^2}} \,dx$$
I am familiar with using the right triangle diagram and theta, but I do not know which terms would go on the hypotenuse and sides in this case. If you can determine which number... | $\int \frac{x}{\sqrt{4-(x+1)^2}}dx = \int \frac{2\sin\theta-1}{\sqrt{4-4\sin^2\theta}}(2\cos\theta)d\theta$
(using the substitution $x+1=2\sin\theta$)
$=\int\frac{2\sin\theta-1}{2\cos\theta}2\cos\theta d\theta$
$= \int (2\sin\theta-1) d\theta$
$=-2\cos\theta-\theta +C$
$=-2\left(\frac{\sqrt{3-2x-x^2}}{2}\right) - \sin^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Puzzle: The number of quadratic equations which are unchanged by squaring their roots is My friend asked me this puzzle:
The number of quadratic equations which are unchanged by squaring their roots is
My answer is: 3
$x^2-(\alpha+\beta)x +\alpha\beta = 0$
where $\alpha$ and $\beta$ be the roots.
case 1:
$\alpha$ = ... | How about just writing: $(x-\alpha)(x-\beta)=(x-\alpha^2)(x-\beta^2)$, so either $\alpha = \alpha^2$ and $\beta=\beta^2$ or $\alpha=\beta^2$ and $\beta=\alpha^2$.
If $\alpha=\alpha^2$, then $\alpha=0\text{ or }1$. Similarly, $\beta=0\text{ or } 1$. So there are three such equations (because $(\alpha,\beta)=(0,1)$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 1
} |
How to find the upper bound for this series I came across this series in a book and the author tells the upper bound and it checks out under the conditions but i could n't find the connection or the reason if u may, that had i not been presented with this info, that i could have used to find the upper bound myself.
For... | Use the fact that if $|x|<1$, then $\frac{1}{1-x}$ has the power series expansion
$$\frac{1}{1-x}=1+x+x^2+x^3+\cdots +x^n+\cdots .$$
Or, in less fancy language, if $|x|<1$, then the infinite geometric series $1+x+x^2+\cdots+x^n+\cdots$ has sum $\frac{1}{1-x}$. In your particular example, $x=\frac{1}{2^{s-1}}$. Since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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What is the least value of the function $y= (x-2) (x-4)^2 (x-6) + 6$? What is the least value of function:
$$y= (x-2) (x-4)^2 (x-6) + 6$$
For real values of $x$ ?
Does $\frac{dy}{dx} = 0$, give the value of $x$ which will give least value of $y$?
Thanks in advance.
| Note that $(x-2)(x-6)=x^2-8x+12$ and $(x-4)^2=x^2-8x+16$. This suggests the symmetrizing substitution $w=x^2-8x+14$. Thus
$$y=(w-2)(w+2)+6=w^2+2.$$
We want to minimize the absolute value of $w$. But $w=(x-4)^2-2$. So $w$ has minimum absolute value $0$, reached when $(x-4)^2=2$, that is, when $x=4\pm\sqrt{2}$. The mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square? Are there any positive $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?
I tried to simplify
\begin{align*}
n^4+n^3+n^2+n+1 &= n^2(n^2+1)+n(n^2+1)+1\\
&= (n^2+n)(n^2+1)+1 \\
&= n(n+1)(n^2+1)+1
\end{align*}
Then I assumed that the above expressi... | Assuming you want positive integers $n$,
I believe we can show that
$$(2n^2 + n)^2 \lt 4(n^4 + n^3 + n^2 + n + 1) \lt (2n^2 + n + 1)^2$$
for $n \gt 3$.
Note: A similar inequality can be given for negative $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/116064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$
If $|z|=1$, show that: $$\mathrm{Re}\left(\frac{1 - z}{1 + z}\right) = 0$$
I reasoned that for $z = x + iy$, $\sqrt{x^2 + y^2} = 1\implies x^2 + y ^2 = 1$ and figured the real part would be:
$$\frac{1 - x}{1 + x}$$
I tried a number of mani... | Not as elegant as many other approaches, but one I thought I'd try out:
Taking the ratio $ \ \frac{1 - z}{1 + z } \ = \ w \ = \ \zeta + i·\eta \ $ to be equal to some complex number, we may arrange the equation into $ \ [ \ 1 - (\cos \theta + i·\sin \theta) \ ] \ = \ [ \ 1 + (\cos \theta + i·\sin \theta) \ ] · (\zeta ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 9,
"answer_id": 8
} |
Solve $ x^2+4=y^d$ in integers with $d\ge 3$
Find all triples of integers $(x,y,d)$ with $d\ge 3$ such that $x^2+4=y^d$.
I did some advance in the problem with Gaussian integers but still can't finish it. The problem is similar to Catalan's conjecture.
NOTE: You can suppose that $d$ is a prime.
Source: My head
| What follows only takes care of $d=3$. We use the Gaussian integer approach that you mentioned. The method can be used with other $d$, but some details of the calculation depend on $d$. Thus one can only deal with one $d$ at a time.
We first examine the case $x$ odd. Factor $x^2+4$ in the Gaussian integers as $(x+2i)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
} |
Are there exact expressions for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$? I was just wondering if there is any way to get an exact expression (with radicals) for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$. In case it's relevant, I want to express $z = \sqrt[4]{8} e^{\frac{5\pi}{8}i}$ in binomial form, and I... | Yes, we can use the double angle formula
$$ 2\cos^2 x - 1 = \cos 2x$$
Pick $x = \frac{3\pi}{8}$ and you can solve it.
Once you have the cos value, you can easily get the sin value.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/119530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Probability for roots of quadratic equation to be real, with coefficients being dice rolls. I really need help with this question.
The coefficients $a,b,c$ of the quadratic equation $ax^2+bx+c=0$ are determined by throwing $3$ dice and reading off the value shown on the uppermost face of each die, so that the first di... | For roots to be real,
$ b^2 - 4ac >= 0 $
the following values of {b,a,c} are possible,
{2,1,1}
{3,1,1} {3,1,2} {3,2,1}
{4,1,1} {4,1,2} {4,2,1} {4,2,2} {4,3,1} {4,13,}
{5,1,1} {5,1,2} {5,2,1} {5,2,2} {5,3,2} {5,2,3} {5,3,1} {5,1,3}
{6,1,1} {6,1,2} {6,2,1} {6,2,2} {6,3,1} {6,1,3} {6,3,3} {6,3,2} {6,2,3}
These are 27 ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Determine the average value of the following function Please bear with me because I have only little experience in using codes to construct the symbols for the equations.
The question is:
Determine the average value of $f(x,y) = x^2 y^2$, in the region $$R: a\le x\le b, c\le y\le d,$$
where $a+b=5, ab=13, c+d=4, cd=7.... | I guess this example could be a weird way to show possible transformation formalities during (double-)integration. Let's just do it formally:
$$\dfrac{\displaystyle\iint f(x,y)dA}{(b-a)(d-c)} = \frac{\int\limits_a^b\int\limits_c^d x^2y^2 dxdy}{(b-a)(d-c)} = \frac{\int\limits_a^b x^2 dx}{b-a}\times\frac{\int\limits_c^d ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to expand $\cos nx$ with $\cos x$? Multiple Angle Identities:
How to expand $\cos nx$ with $\cos x$, such as
$$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$
See a list of trigonometric identities in english/ chinese
| There were two corrections in the posting ($1102$ should be $1120$ and $\cos x$ term should be $\cos^2 x$ term:
$$\cos 10x = 512 (\cos x)^{10} -1280 (\cos x)^8 +1120 (\cos x)^6 -400(\cos x)^4+50(\cos x)^2-1$$
Follow Arturo's answer you should get the following
$$
\begin{align*}
\cos 2x &= 2 \cos^2 x -1 \\
\cos 3x &=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 1
} |
Partial fraction with a constant as numerator I am trying to express this as partial fraction:
$$\frac{1}{(x+1)(x^2+2x+2)}$$
I have a similar exaple that has $5x$ as numerator, it is easy to understand. I do not know what to do with 1 in the numerator, how to solve it?!
| You want to find $A, B,$ and $C$ such that
$$\frac{1}{(x+1)(x^2 + 2x + 2)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 2x + 2}
$$
That is such that
$$\begin{align}0x^2 + 0x + 1 &= A(x^2 + 2x + 2) + (x+1)(Bx+c)\\
&= (A+B)x^2 + (2A+B+C)x + 2A + C.
\end{align}$$
So you get three equations
$$\begin{align} 0 &= A + B \\
0 &=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$? Assume $p$ is prime and $p\ge 3$.
Through experimentation, I can see that it's probably true. Using Wilson's theorem and Fermat's little theorem, it's equivalent to saying $2^2 4^2 6^2 \cdots (p-1)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$, but I can't figure out an... | $$(p-1)!=1\cdot2\cdots\frac{p-1}{2}\cdot\frac{p+1}{2}\cdots(p-2)(p-1)$$ We have the congruences
$$\begin{align*}
p-1&\equiv -1\pmod p\\
p-2&\equiv -2\pmod p\\
&\vdots\\
\frac{p+1}{2}&\equiv -\frac{p-1}{2}\pmod{p}\end{align*}$$
Rearranging the factors produces
$$(p-1)!\equiv 1\cdot(-1)\cdot2\cdot(-2)\cdots\frac{p-1}{2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Meaning of this 4x4 determinant Let $p,q,r$ and $s$ be four points on the plane. Moreover, $p,q,r$ are given in clockwise order. My book said that the following determinant is positive if and only if $s$ lies inside the circle passing through $p,q,r$. Why?
$$\det
\begin{bmatrix}
p_x & p_y & p_x^2+p_y^2 & 1 \\
... | Following the hints from J.M., I was able to get the answer.
$$\det
\begin{bmatrix}
p_x & p_y & p_x^2+p_y^2 & 1 \\
q_x & q_y & q_x^2+q_y^2 & 1 \\
r_x & r_y & r_x^2+r_y^2 & 1 \\
s_x & s_y & s_x^2+s_y^2 & 1 \\
\end{bmatrix}
$$
$$
=-a(s_x^2+s_y^2)-bs_x+cs_y+d\\
=-a(s_x^2+s_y^2+\frac{b}{a}s_x-\frac{c}{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How to find perpendicular vector to another vector? How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$
Could anyone explain this to me, please?
I have a solution to this when I have $3\mathbf{i}+4\mathbf{j}$, but could not solve if I have $3$ components...
When I google... | One way to do this is to express the vector in terms of a spherical coordinate system. For example
$$ \boldsymbol{e}= \pmatrix{a \\ b \\ c} = r \pmatrix{ \cos\varphi \cos\psi \\ \sin\varphi \cos\psi \\ \sin\psi} $$
where $r=\sqrt{a^2+b^2+c^2}$, $\tan(\varphi) = \frac{b}{a}$ and $\tan{\psi} = \frac{c}{\sqrt{a^2+b^2}}.$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "78",
"answer_count": 18,
"answer_id": 6
} |
How Can One Prove $\cos(\pi/7) + \cos(3 \pi/7) + \cos(5 \pi/7) = 1/2$ Reference: http://xkcd.com/1047/
We tried various different trigonometric identities. Still no luck.
Geometric interpretation would be also welcome.
EDIT: Very good answers, I'm clearly impressed. I followed all the answers and they work! I can only... | To elaborate on Mathlover's comment, the three numbers $\cos\frac{\pi}{7}$, $\cos\frac{3\pi}{7}$, and $\cos\frac{5\pi}{7}$ are the three roots of the monic Chebyshev polynomial of the third kind
$$\hat{V}_n(x)=\frac{\cos\left(\left(n+\frac12\right)\arccos\,x\right)}{2^n\cos\frac{\arccos\,x}{2}}=\frac1{2^n}\left(U_n(x)-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/140388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 8
} |
How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$ How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$
Edit: I'm specifically stuck on showing that $\frac{2^n - 1}{2^n} + \fra... | Just multiply the numerator and denominator in the first fraction by 2 and add the fractions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/141126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Simplifying Logarithmic Expression
Compute:
$$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$
I tried to expand it :
$$\frac{1-\log_a^{3}{b} }{(\log_a b+\log_b a+1)\log_a\frac{a}{b}}$$
$$=\frac{(1-\log_a{b})(\log_a^{2}b+\log_a b+1)}{(\log_a b+\log_b a+1)(1-\log_a{b})}$$
$$=\frac{(\log_a^{2}b+\log_a ... | $$ \large{x = \log_a b}$$
Then $$ 1- \log_a^{3}{b} = (1-x^3) $$
Also
$$(\log_a b+\log_b a+1)(\log_a\frac{a}{b}) = (x+\frac{1}{x}+1)(1-x)$$
because
$$ \log_b a = \frac{1}{\log_a b} \hspace{8pt} \textit{and} \hspace{8pt} \log_a \frac{a}{b} = (1-\log_a b)$$
The whole thing gets simplified to
$$ \frac{(1-x^3)}{(x+\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/142197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How to find $\int{\frac{\sqrt{x^2+1}}{x+2}dx}$ I have got such integral $$\int{\frac{\sqrt{x^2+1}}{x+2}dx}$$ and with Maple I got something like this:
$$\int\frac{1}{2} + \frac{1+3u^2+4u^3}{-2u^2+2u^4-8u^3}du$$ And I want to know how to achive this changes.
I tried to use WolframAlpha, but there is scarier solution. Th... | \begin{align}
&\int{\frac{\sqrt{x^2+1}}{x+2}dx}\\
= &\int \frac{{(x^2-4)+5}}{(x+2)\sqrt{x^2+1}}dx
= \int \frac{x-2}{\sqrt{x^2+1}} +\frac{5}{(x+2)\sqrt{x^2+1}}\ dx\\
=&\ \sqrt{x^2+1}-2\sinh^{-1}x+\sqrt5\tanh^{-1}\frac{2x-1}{\sqrt{5(x^2+1)}}+C
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/145066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding the integral of $x/\sqrt{4-x^2}$ Find the integral:
$$\int \frac{x}{\sqrt{4-x^2}} dx = \int \frac{x}{\sqrt{2^2-x^2}} dx$$
using $$\int \frac1{\sqrt{a^2-x^2}} dx = \arcsin(x/a) + C$$
I get $\displaystyle \frac{x^2}{2} \arcsin \left(\frac{x}{2} \right) + C$.
I'm not sure if the $\dfrac{x^2}{2}$ is right. Any sug... | Another way: $$\int\frac{x}{\sqrt{4-x^2}}dx=-\frac{1}{2}\int\frac{d(4-x^2)}{(4-x^2)^{1/2}}dx=-\frac{1}{2}\frac{\sqrt{4-x^2}}{1/2}+C=-\sqrt{4-x^2}+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/146543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Does $\int_0^\infty\frac{\cos^2x}{x^2+5x+11}dx$ converge or diverge? When I'm learning convergence, my teacher just show me about condition to convergence or not. But I haven't meet a function that contain both trigonometric and normal polynomial. When I asked one of my friends, he tell me that using Taylor to develope... | Let $$I = \int_0^{\infty} \frac{\cos^2(x)}{x^2 + 5x+11} dx$$ The integrand is non-negative and since $\cos^2(x) \in [0,1]$, $\forall x \in \mathbb{R}$, we get that $$0 \leq I = \int_0^{\infty} \frac{\cos^2(x)}{x^2 + 5x+11} dx \leq \int_0^{\infty} \frac1{x^2 + 5x+11} dx = \int_0^{\infty} \frac{dx}{(x+5/2)^2 + 19/4}$$
No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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What equation intersects only once with $f(x)=\sqrt{1-(x-2)^2}$ Being $f(x)=\sqrt{1-(x-2)^2}$ I have to know what linear equation only touches the circle once(only one intersection), and passes by $P(0,0)$.
So the linear equation must be $y=mx$ because $n=0$.
I have a system of 2 equations:
\begin{align}
y&=\sqrt{1-(x-... | As you have in your post, we have $y = mx$ as the straight line. For this line to touch the semi-circle, we need that $y = mx$ and $y = \sqrt{1 - (x-2)^2}$ must have only one solution. This means that the equation $$mx = \sqrt{1 - (x-2)^2}$$ must have only one solution.
Hence, we need to find $m$ such that $m^2x^2 = 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147874",
"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.