Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Test the convergence of $\sum_{n=1}^{n=\infty}\left(\frac{n^2}{2^n}+\frac{1}{n^2}\right)$. Test the convergence of
$$
\sum_{n=1}^{\infty}\left(\frac{n^2}{2^n}+\frac{1}{n^2}\right)
$$
$$
\frac{u_{n+1}}{u_{n}}=\frac{\frac{(n+1)^2}{2^{n+1}}+\frac{1}{(n+1)^2}}{\frac{n^2}{2^{n}}+\frac{1}{n^2}}
$$
$$
\lim_{n\to\infty}\frac{... | Note if the two series $\displaystyle \sum_{n=1}^\infty u_n$ and $\displaystyle \sum_{n=1}^\infty v_n$ are convergent then the series $\displaystyle \sum_{n=1}^\infty (u_n+v_n)$ is also convergent.
Now, we know that the Riemann series $\displaystyle \sum_{n=1}^\infty\frac{1}{n^2} $ is convergent and we have
$$\frac{n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$ Is there an "easy" way to find the minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$ without the help of any computer programme?
If I knew $\sin(\pi/8)=\frac{\sqrt{2-\sqrt2}}{2}$ then it would obviously be easy to find it, but how would i evaluate $\sin(\pi/8)... | This is an old question, but it doesn't seem to have a correct answer with full details.
Let $\omega = e^{i\pi/9}$ and $\zeta = e^{i\pi/8}$, so $\cos(\pi/9) = \mathrm{Re}(\omega)$ and $\sin(\pi/8) = \mathrm{Im}(\zeta)$, and note that $\omega$ is a primitive 18th root of unity and $\zeta$ is a primitive 16th root of uni... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$ It is known that$$\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$$
for exactly one set of positive integers $(a,b,c)$ where $0<c<90$
find the value
$$\dfrac{b+c}{a}$$
my idea,$ \sin 50^\circ >\sin 45^\circ >\frac{_5}{^8} $
so$\sqrt{9-8\sin 50^{\cir... | We have the following (working in degrees):
$$\cos 20 - \cos 80 = \cos(50-30) - \cos(50+30) = 2 \sin 50 \sin 30 = \sin 50$$
Thus we have that
$$1 - 2\sin^2 10 - \sin 10 = \sin 50$$
(using $\cos 20 = 1 - 2 \sin^2 10$ and $\cos 80 = \sin (90 - 80) = \sin 10$)
And so
$$9 - 8 \sin 50 = 9 - 8(1 - 2\sin^2 10 - \sin 10) = 1 +... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$ Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$
So if i apply ratio test, I get $\lim_{x\to \infty} 5|x||\frac{n+2^n}{n+1+2^{n+1}}| $
Now need help to to check endpoints x= 2/5 and -2/5
If i pu... | By ratio test
$$ 5|x||\frac{n+2^n}{n+1+2^{n+1}}|\sim_\infty 5|x||\frac{2^n}{2^{n+1}}|=\frac{5}{2}|x|<1\iff|x|<\frac{2}{5}$$
hence the radius is $\frac{2}{5}$.
Added
-For $x=\frac{2}{5}$, $$\frac{5^n}{n+2^n}x^n=\frac{2^n}{n+2^n}\to1\neq0$$
so the series is divergent.
-For $x=-\frac{2}{5}$,
$$\frac{5^n}{n+2^n}x^n=\frac{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $4x-x^4 \leq 3, x \in \Bbb R$ How can I tackle the following inequality :
Prove that $4x-x^4 \leq 3$, where $x$ is any real number.
Can someone point me in the right direction?
| We see that , $x^4-4x+3=x^4-2x^2+1+2x^2-4x+2=(x^2-1)^2+2(x-1)^2 \geq 0$ and so $-(x^4-4x+3)\leq 0 \implies -x^4+4x-3 \leq 0$ and
hence $4x-x^4 \leq 3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
} |
To find the logarithm of $1728$ to the base $2 \sqrt{3}$
Find the logarithm of: $1728$ to base $2\sqrt{3}$.
Let, $\log_{2\sqrt{3}} 1728 = y$, then
$$\begin{align} (2\sqrt{3})^y &= 1728\\
2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= 1728\\2^y(3^\frac y2)
&= 1728\\2^y × 3^\frac y2 &= 2^6 × 3^3 \end{align}$$
What should... | Hint: by the very definition of logarithm
$$\log_{2\sqrt 3}1728=y\iff (2\sqrt 3)^y=1728=12^3=2^6\cdot 36\ldots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Expressing $\cos\theta - \sqrt{3}\sin\theta = r\sin(\theta - \alpha)$ My book explains that $a\cos\theta + b\sin\theta$ is a sine (or cosine) graph with a particular amplitude/shift (i.e. $r\sin(\theta + \alpha)$) and shows me some steps to solve for $r$ and $\alpha$:
$$r\sin(\theta + \alpha) \equiv a\cos\theta + b\sin... | The answer is that it depends on the choice of $\alpha$. This is because $\sin (x + \pi) = - \sin x$ (also if you were using $\cos$ instead, $\cos (x + \pi) = - \cos x$). This means that it could be either positive or negative, but then $\alpha$ may need to be reduced by $\pi$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/363222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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The smallest three digit number that is equal to the sum of its digits plus twice the product of its digits How can I do the following?
Find the least three-digits number that is equal to the sum of its digits plus twice the product of its digit?
| You are looking for integers $a$, $b$, and $c$, with $0\leq a,b,c\leq 9$ ($c\neq 0$), and
$$
a+10b+100c=a+b+c+2abc
$$
Simplifying, that becomes
$$
9b+99c=2abc
$$
We look for the smallest solution, so we want the smallest integer $c$ satisfying
$$
c=\frac{9b}{2ab-99}
$$
From this, we need $2ab\geq100$ or $ab\geq 50$. So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364153",
"timestamp": "2023-03-29T00:00:00",
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Can someone check the solution to this recurrence relation? Here's the recurrence relation: $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$
Here's the solution:Write:
$$
a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = 4
$$
Define $A(z) = \sum_{n \ge 0} a_n z^n$. If you multiply... | After the correction others have pointed out in your first line, you do the same z-transform/generating function stuff you did before, the only difference is now your two forcing terms are shifted by two.
\begin{align}
a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^{n+2} + (n+2) + 3 \quad a_0 = 1, a_1 = 4
\end{align}
You really o... | {
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"timestamp": "2023-03-29T00:00:00",
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With the binomial expansion of $ (3+x)^4$ express $(3 - \sqrt 2)^4$ in the form of $p+q\sqrt 2$ I know how to expand the binomial expansion but I have no idea how to do this second part.
Hence Express $$(3 - \sqrt 2)^4$$ in the form of $$P+Q\sqrt 2$$
Where P and Q are integers, and then how to state the values of P and... | Using the binomial theorem
$$
\begin{align*}
(3-\sqrt{2})^4 &= \sum_{k = 0}^4 \binom{4}{k} (3)^k (-\sqrt{2})^{4-k}\\
&= 3^0\binom{4}{0}(-\sqrt{2})^4 + (3^1) \binom{4}{1} (-\sqrt{2})^3 + (3^2)\binom{4}{2}(-\sqrt{2})^2 \\
&\quad + (3^3)\binom{4}{3}(-\sqrt{2}) + (3^4)\binom{4}{4}(-\sqrt{2})^0\\
&= (1)(1)(4) + (3)(4)(-2\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the coefficient in the closed form of the generating function I try to solve the recursion $a_n=5a_{n-1}+5^n$ with $a_0=1$ with generating function, but I could not find the coefficient of $x^n$ in the closed form
\begin{eqnarray*}
g(x)&=&a_0+\sum^n_1a_nx^n\\
g(x)&=&1+\sum^n_1(5a_{n-1}+5^n)x^n\\
g(x)&=&1+5\sum^... | You get the generating function:
$$
g(z) = \frac{1}{(1 - 5 z)^2}
= \sum_{n \ge 0} \binom{-2}{n} (-1)^n 5^n z^n
= \sum_{n \ge 0} \binom{n + 2 - 1}{2 - 1} 5^n z^n
= \sum_{n \ge 0} (n + 1) \cdot 5^n z^n
$$
so that:
$$
a_n = (n + 1) \cdot 5^n
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/367820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integer solutions of $x^3+y^3=z^2$ Is there any integer solution other than $(x,y,z)=(1,2,3)$ for $x^3+y^3=z^2$?
| It may be of help to consider that $x^3 + y^3 = ( x + y ) \cdot ( x^2 - xy + y^2)$, so one could start by looking for values of $x$ and $y$ for which those factors are equal (and then for values where one factor "completes a square" with the other).
ADDENDUM: I had a little time to think more on this during my snowy w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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"answer_id": 2
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How can I solve these two equations to find theta? I am doing a projectile motion questions and I have to solve these simultaneous equations:
$$\frac{-5t^{2}+30t\sin\theta }{30t\cos\theta }=\frac{1}{\sqrt3}$$
$$\frac{-10t+30\sin\theta }{30\cos\theta }=-\sqrt3$$
I solved them but the solution is long and tedious and I g... | Subtracting the second from the first equation gives
$$\frac{t}{6\cos\theta} = \frac{4}{3}\sqrt{3}$$
or
$$t=8\sqrt{3}\cos\theta.$$
Plugging this back into the first equation gives
$$ \frac{-4\sqrt{3}\cos\theta+3\sin\theta}{3\cos\theta} = \frac{1}{3}\sqrt{3}$$
or
$$3\sin\theta = 5\sqrt{3}\cos\theta.$$
Thus finally we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable? In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
I would like to know if this resu... | The solutions are of the form $\displaystyle(p, q)= \left(\frac{3t^2nr+n^3}{8},\,\frac{3n^2t+t^3r}{8}\right)$, for any rational parameter $t$. To prove it, we start with $$\left(p+q\sqrt{r}\right)^{1/3}+\left(p-q\sqrt{r}\right)^{1/3}=n\tag{$\left(p,q,n,r\right)\in\mathbb{N}^{4}$}$$
and cube both sides using the identit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
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Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
Let $a_1, a_2, a_3, \dots , a_n$ be positive real numbers whose sum is $1$. Prove that
$$\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \ldots +\frac{a_n^2}{a_n+a_1} \geq \frac12\,.$$
I thought maybe the Cauchy and QM in... | Thanks to Sanchez for giving me a hint to solve this. Here is a full solution.
By the Cauchy-Schwarz inequality we have:
$${\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1}=\frac{a_1^2}{(\sqrt{a_1+a_2})^2}+\frac{a_2^2}{(\sqrt{a_2+a_3})^2}+ \cdots+ \frac{a_n^2}{(\sqrt{a_n+a_1})^2} \geq \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
integrating fraction/completing the square Does anyone know how to integrate the following?
$\frac{dx}{9x^2 + 6x + 17}$
I have been trying for ages and cannot get an answer anywhere close to the answer I get on maple?
| $$9x^2+6x+17=(3x)^2+2\cdot3x\cdot1+1^2+17-1=(3x+1)^2+4^2$$
Put $3x+1=4\tan\theta$ so that $(3x+1)^2+4^2=\cdots=16\sec^2\theta$ and $3dx=4\sec^2\theta d\theta$
So, $$\int \frac{dx}{9x^2+6x+17}=\int\frac{4\sec^2\theta d\theta}{3\cdot16\sec^2\theta}=\frac1{12}\int d\theta=\frac{\theta}{12}+C=\frac{\arctan\left(\frac{3x+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Eigenvector of matrix of equal numbers For matrix the matrix
$$A = \begin{bmatrix}
3&1&1\\
1&3&1\\
1&1&3\\
\end{bmatrix}$$
with eigenvalues $\lambda_1=5$, $\lambda_2=2$, $\lambda_3=2$, I am trying to find the corresponding eigenvector corresponding to the eigenvalue 2. I got
$$(A - 2I_3) = \begin{bmatrix}
1&1&1\\
1&1&1... | For any square matrix with one value on the diagonal and another value everywhere else, a consistent pattern of (orthogonal) eigenvectors for the $n$ by $n$ case can be read from the columns of
$$
\left( \begin{array}{rrrrrrrrrr}
1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\
1 & 1 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How to count permutations with restrictions on how items are grouped I am trying to solve the following problem:
A town contains $4$ people who repair televisions. If $4$ sets break down, what is the probability that exactly $i$ of the repairers are called? Solve the problem for $i=1,2,3,4$.
For $i=1$, there are ${}_... | Conditions:
*
*The repairmen and TVs are distinct.
*Every TV owner calls ONLY one repairman, that is, the relationship between the set of TVs, $T$, and a set that contains $i$ of the repairmen, $Ri$, is functional, $f: T -> Ri$. Note that $|T|=4$ and $|Ri|=i$
*Every one of the $i$ repairmen gets at least one call,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Integral Question - $\int\frac{1}{\sqrt{x^2-x}}\,\mathrm dx$ Integral Question - $\displaystyle\int\frac{1}{\sqrt{x^2-x}}\,\mathrm dx$.
$$\int\frac{1}{\sqrt{x(x-1)}}\,\mathrm dx =\int \left(\frac{A}{\sqrt x} + \frac{B}{\sqrt{x-1}}\right)\,\mathrm dx$$
This is the right way to solve it?
Thanks!
| The Partial Fraction Decomposition is for rational fraction only.
$$\int\frac{dx}{\sqrt{x^2-x}}=\int\frac{2dx}{\sqrt{4x^2-4x}}=\int\frac{2dx}{\sqrt{(2x-1)^2-1^2}}$$
Now, put $2x-1=\sec\theta$
EDIT: completing as requested
So,$2dx=\sec\theta\tan\theta d\theta$
$$\text{So,}\int\frac{2dx}{\sqrt{(2x-1)^2-1^2}}=\int \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Is there a pattern in figures whose perimeter is the same as their area? Here's what I've found so far.
*
*Circumference of a circle with identical circumference and area: $4\pi$
*Side length of a triangle with identical perimeter and area: $4\sqrt{3}$
And so on...
*
*Square: $4$
*Pentagon: $4\sqrt{5 - 2\sqrt... | We can look at a regular $n$-gon of "radius" $r$, i.e., the convex hull of $r$ times the $n$-th roots of unity.
Connecting the vertices of the polygon to the origin gives you $n$ isosceles triangles of area $\frac{r^2}{2}\sin \frac{2\pi}{n}$. The total area of the $n$-gon is thus
$$A = \frac{nr^2}{2} \sin \frac{2\pi}{n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ Inadvertently, I find this interesting inequality. But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find this
$$\ln{2}<\left(\dfrac{1}{2}\right)^{\frac{1}{2}}=\dfrac{\sqrt{2}}{2... | Hmm, first I thought the following would allow to compute the proof mentally, but, well... although I find it a remarkable simplification I'll need the pocket calculator in the end. But let's see:
$$\ln(2) \gt (2 / 5)^{2 /5} = \left({16 \over 100 }\right)^{1/5} $$ We have also $$
\ln(2) = \ln \left( 1+1/3 \over ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "95",
"answer_count": 8,
"answer_id": 5
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On the lengths of the cycles of random permutations A random permutation of $n$ from $n$ is chosen. What is the mathematical expectation of the sum of squares of lengths of its cycles?
| The exponential generating function of permutations by the sum of the squares of the lengths of its cycles is
$$ G(z, u) =
\exp
\left(u z + \frac{1}{2} u^4 z^2 + \frac{1}{3} u^9 z^3 + \frac{1}{4} u^{16} z^4 \cdots\right)$$
which means that the probability generating function of the expectation that we are looking for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral of $\sin x \cdot \cos x$ I've found 3 different solutions of this integral. Where did I make mistakes? In case there is no errors, could you explain why the results are different?
$ \int \sin x \cos x dx $
1) via subsitution $ u = \sin x $
$ u = \sin x; du = \cos x dx \Rightarrow \int udu = \frac12 u^2 \Righta... | $$\frac{d\{f(x)+c\}}{dx}=f'(x)$$ for any arbitrary constant $c$
$$\implies \int f'(x)dx=f(x)+d $$ for any arbitrary constant $d$
So, in indefinite integral we can get answers which differ by some constant
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/381243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Integral Of $\int\sqrt{\frac{x}{x+1}}dx$ I want to solve this integral
$$\int\sqrt{\frac{x}{x+1}}dx$$
And think about:
1) $t=\frac{x}{x+1}$
2) $dt = (\frac{1}{x+1} - \frac{x}{(x+1)^2})dx$
Now I need your advice! Thanks!
| If you know about Möbius transformations, from the change of variables
$$
t = \frac{x}{x+1}, \quad \Longrightarrow \quad x = \frac{t}{1-t}
$$
because
$$
\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}^{-1} = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}.
$$
If you don't know about Möbius transformations, then you can do t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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How can I evaluate this given improper integral? How can I evaluate this integral:
$$\int _{ 0 }^{3}{ \frac { x }{ (3-x)^{\frac{1}{3}}} dx} \ ?$$
| Try $y = (3-x)^{1/3}$.
$y^3 = 3-x$,
so $x = 3-y^3$
and $dx = -3 y^2 dy$.
Putting this in,
since $y$ goes from $3^{1/3}$ to $0$
as $x$ goes from $0$ to $3$,
$\begin{align}
\int_{ 0 }^{3}{ \frac { x }{ (3-x)^{\frac{1}{3}}} dx}
&=\int_{3^{1/3}}^0 \frac{3-y^3}{y}(-3 y^2)dy\\
&=\int_0^{3^{1/3}} \frac{3-y^3}{y}(3 y^2)dy\\
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/386291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$ Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
| For $x < 1$, we have the Taylor series expansion:
$$f(x):= \frac{-1}{4} \log \left(- \frac{x - x^{-1}}{x + x^{-1}} \right) = \frac{x^2}{2} + \frac{x^6}{6} + \frac{x^{10}}{10} + \frac{x^{14}}{14} + \ldots $$
Then
$$f(x) + \frac{f(x^2)}{2} + \frac{f(x^4)}{4} + \frac{f(x^8)}{8} + \ldots
= \frac{x^2}{2} + \frac{x^4}{4} + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 2,
"answer_id": 0
} |
How can this $T(n) = T(n-1)+T(n-2)+3n+1$ non homogenous recurrence relation be solved How are can the above recurrence relation be solved?
I've reached here:
$(x^{2}-x-1)(x-3)^2(x-1)$
And then here:
$$a_n = l_1 \cdot (x_1)^n+l_2 \cdot (x_2)^n+l_3 \cdot (x_3)^n+l_4\cdot n \cdot (x_3)^n+l_5\cdot (x_4)^n$$
And we are giv... | Use "generatingfunctionology" techniques. Define $G(z) = \sum_{n \ge 0} T(n) z^n$, write your recurrence as:
$$
T(n + 2) = T(n + 1) + T(n) + 3 n + 3
$$
Multiply by $z^n$, add for $n \ge 0$. Remember that:
$$
\sum_{n \ge 0} (n + 1) z^n = \frac{1}{(1 - z)^2}
$$
and you get:
$$
\frac{(G(z) - T(0) - T(1) z}{z^2}
= \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
| HINT:
Using Partial Fraction Decomposition formula,
$$\frac{x^4+2x+4}{x^4-1}=1+\frac{ax+b}{x^2+1}+\frac c{x+1}+\frac d{x-1}$$ where $a,b,c,d$ are arbitrary constants to determined by equating the coefficients of the different powers of $x$ in
$$x^4+2x+4=x^4-1+(x^2-1)(ax+b)+c(x-1)(x^2+1)+d(x+1)(x^2+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/391485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$A_2
\begin{pmatrix} 1& 1\\
1& 2
\end{pmatrix};
A_3 = \begin{pmatrix} 1& 1& 1\\
1& 2& 2\\
1& 2& 3
\end{pmatrix};
A_4 = \begin{pmatrix} ... | Recall that adding a multiple or subtracting a multiple of one row does not change the value of the determinant, see, for example ProofWiki.
Using this fact and Laplace expansion you get
$$|A_4|=
\begin{vmatrix}
1& 1& 1& 1\\
1& 2& 2& 2\\
1& 2& 3& 3\\
1& 2& 3& 4
\end{vmatrix}=
\begin{vmatrix}
1& 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$? Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
| $$
\begin{align}
n\cot^{-1}(n)-1
&=n\tan^{-1}\left(\frac1n\right)-1\\
&=n\int_0^{1/n}\frac{\mathrm{d}x}{1+x^2}-1\\
&=-n\int_0^{1/n}\frac{x^2\,\mathrm{d}x}{1+x^2}\\
&=-\int_0^1\frac{x^2\,\mathrm{d}x}{n^2+x^2}\tag{1}
\end{align}
$$
Using formula $(9)$ from this answer and substituting $z\mapsto ix$, we get
$$
\sum_{n=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/393013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 3,
"answer_id": 1
} |
Integrate by parts: $\int \ln (2x + 1) \, dx$ $$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x + 1) \, dx = x\ln (2x + 1) - \int {2x \over 2x + 1} \cr
& = x\ln (2x + 1) - \int 1 - {1 \over... | I am sure there are more tidy ways to do this but as an alternative...
Why not do $\int \ln(2x + 1)dx$ using:
$v^\prime = 1 \Rightarrow v = x$ and $u = \ln(2x+1)\Rightarrow u^\prime=\frac{2}{2x+1}$
Therefore,
$$\int \ln(2x + 1)dx = x\ln(2x+1) - \int \frac{2x}{2x+1}dx$$
Then make the substitution $u=2x + 1$ to yield,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/393929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Range of a function Find the range of
$$f(x)=\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$$ where $a, b, c$ are distinct real numbers such $a\neq b\neq c\neq a$.
| $$f(x)=\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$$
$$f(a)= \dfrac{(a-b)(a-c)}{(a-b)(a-c)}=1$$
$$f(b)= \frac{(b-c)(b-a)}{(b-c)(b-a)}=1$$
$$f(c)= \frac{(c-a)(c-b)}{(c-a)(c-b)}=1$$
Now what is $f'(x)?$, $f'(x)=\dfrac{2x-(a+b)}{(c-a)(c-b)}+\dfrac{2x-(b+c)}{(a-b)(a-c)}+\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/395381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
real mechanism behind addition, subtraction, multiplication, division we all know the basic rules for operations of addition, subtraction, multiplication and division.
but what i don't know is why these rules (of addition, subtraction, multiplication and division) works.
as if we have been given algorithm to do these ... | $$
\begin{array}{cccccc}
& & 4 & 3 \\
& \times & 7 & 9 \\
\hline & 3 & 8 & 7 \\
3 & 0 & 1 \\
\hline 3 & 3 & 9 & 7
\end{array}
$$
Why is this algorithm, taught in elementary school, justified?
It's because multiplication distributes over addition:
$$
9\cdot(43) = 9\cdot(4\cdot10 \quad+\quad 3) = (9\cdot4)(10)\quad+\qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Generating function: Find a closed form of $\sum_{k=0}^n (-3)^k(k+1)$ Find the closed form of $\sum_{k=0}^n (-3)^k(k+1)$.
So the generating function would be: $$A(x)=1-6x+18x^2-108x^3...$$ So what I did notice is that its closed form is perhaps some variation of $1\over {1+x}$ but I didn't manage to find a general form... | Let $f(x) = \sum_{k=0}^n x^{k+1}
= \frac{x-x^{n+2}}{1-x}
$.
Then $f'(x) = \sum_{k=0}^n (k+1)x^k$,
so $f'(-3) = \sum_{k=0}^n (k+1)(-3)^k$.
So
$\begin{align}
f'(x) &= \frac{(x-x^{n+2})'(1-x) - (1-x)'(x-x^{n+2})}{(1-x)^2}\\
&= \frac{(1-x)(1-(n+2)x^{n+1})+x-x^{n+2}}{(1-x)^2}\\
&= \frac{1-(n+2)x^{n+1}-x+(n+2)x^{n+2}+x-x^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/397791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
is $(x+1)^4-x^4$ non-prime for all natural positive integers $x$ Looking at difference between two neighbouring positive integers raised to the power 4, I found that all differences for integer neighbours up to $(999,1000)$ are non-prime.
Does this goes for all positive integers?
And can someone please prove?
| Hint : you can write $(x+1)^4-x^4=((x+1)^2-x^2)((x+1)^2+x^2)=(x+1-x)(x+1+x)((x+1)^2+x^2)=(2x+1)((x+1)^2+x^2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/399149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$ Find the limit without the use of L'Hôpital's rule or Taylor series
$$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$
| sorry about to answer my Question but Zarrax's way and her comment lead me to the answer
$$L=\lim_{x\rightarrow 0}\frac{\sin(x)-x}{x^3}=\lim_{x\rightarrow 0}\frac{\sin\frac{x}{3}-\frac{x}{3}}{(\frac{x}{3})^3}$$
$$L=\lim_{x\rightarrow 0}\frac{\sin(x)-x}{x^3}=\lim_{x\rightarrow 0}\frac{3\sin(\frac{x}{3})-4\sin^3(\frac{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
Finding the limit, multiplication by the conjugate I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$
I tried and tried... friends of mine tried as well and we don't know how to get out of:
$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$
(this is what we get after multiplying by the conjugate of $2 + \sqrt{3+x... | You had the right idea: the issue is in your simplification of the numerator:
$$\begin{align}
(2 - \sqrt{3 + x})(2 + \sqrt{3 + x}) & = 2^2 - \left(\sqrt{(3 + x)}\right)^2 \\ \\
& = 4 - (3 + x) \\ \\
& = 4 - 3 - x \\ \\
& = 1 - x = -(x - 1)
\end{align}$$
That gives you $$\begin{align} \lim_{x \to 1} \frac{-(x - 1)}{(x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Telescoping sum of powers
$$
\begin{array}{rclll}
n^3-(n-1)^3 &= &3n^2 &-3n &+1\\
(n-1)^3-(n-2)^3 &= &3(n-1)^2 &-3(n-1) &+1\\
(n-2)^3-(n-3)^3 &= &3(n-2)^2 &-3(n-2) &+1\\
\vdots &=& &\vdots & \\
3^3-2^3 &= &3(3^2) &-3(3) &+1\\
2^3-1^3 &= &3(2^2) &-3(2) &+1\\
1^3-0^3 &= &3(1^2) &-3(1) &+1\\
\underline{\hphantom{(... | Look at $n^3 - (n-1)^3 = 3n^2 - 3n + 1$. Now substitute.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Finding the fraction $\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}$ when knowing the sums $a+b+c+d$ to $a^4+b^4+c^4+d^4$ How can I solve this question with out find a,b,c,d
$$a+b+c+d=2$$
$$a^2+b^2+c^2+d^2=30$$
$$a^3+b^3+c^3+d^3=44$$
$$a^4+b^4+c^4+d^4=354$$
so :$$\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}=?$$
If the qusetion i... | I do not know how to solve it, but $a=-3,b=4,c=2,d=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 2
} |
curl of what yields $(0,s^{-1},0)$ in cylindrical coordinates? In cylindrical coordinates $(s,\theta,z)$, what function $\mathbf{A}$ has the property $$\nabla\times \mathbf{A} = (0, \frac{1}{s} , 0) $$
I know generally that $$\nabla\times \mathbf{A} = \left(\frac{1}{s}\frac{\partial A_3}{\partial \theta}-\frac{\partial... | There is a path integral formula to construct the inverse of the exterior derivative operator for closed differential forms, for curl in three dimensions:
$$
\newcommand{\A}{\mathbf{A}}\newcommand{\ps}{\boldsymbol{\psi}} \newcommand{\p}{\mathbf{p}} \mathscr{R}(\ps) = -(\p - \p_0)\times \int^1_0 \ps\big(\p_0 + t(\p- \p_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Elementary Row Matrices Let $A$ =
$$
\begin{align}
\begin{bmatrix}
-4 & 3\\
1 & 0
\end{bmatrix}
\end{align}
$$
Find $2 \times 2$ elementary matrices $E_1$,$E_2$,$E_3$ such that $A$ = $E_1 E_2 E_3$
I figured out the operations which need to be performed which are;
$E_1$ = $R_2 \leftrightarrow R_1$
$E_2$ = $R_2$ = $R_2$... | Note that the solutions are not unique. With your elementary row operations, we have
$$
\pmatrix{1&0\\ 0&\tfrac13}\pmatrix{1&0\\ 4&1}\pmatrix{0&1\\ 1&0}A = I_2.
$$
Therefore, by performing the reverse row operations (and also in reverse order) on $I_2$, we get
$$
A = \pmatrix{0&1\\ 1&0}\pmatrix{1&0\\ -4&1}\pmatrix{1&0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$,
where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$.
Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ where $s_k$ satisfies the recurrence relation
\begin{align}
& s_{... | let $$y=\sum_{n=1}^{\infty}H^2_{n}x^n$$
then we have
$$y=x+xy+\ln^2{(1-x)}+\int_{0}^{x}\dfrac{\ln{(1-t)}}{t}dt$$
so
$$y=\dfrac{\ln^2{(1-x)}}{1-x}+\sum_{n=1}^{\infty}\left(1+\dfrac{1}{2^2}+\cdots+\dfrac{1}{n^2}\right)x^n$$
then you can use:Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/405356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 0
} |
Simple divisibility problem
Suppose $a$ and $b$ are distinct integers and $n\mid a^n-b^n$. Prove that $n\mid\frac{a^n-b^n}{a-b}$.
Here is how I do it: Write $a=b+d$, then $a^n-b^n=(b+d)^n-b^n=\binom{n}{1}b^{n-1}d+\binom{n}{2}b^{n-2}d^2+ \cdots + \binom{n}{n-1}bd^{n-1}+d^n \equiv 0 \pmod{n}$. Since $n| \binom{n}{k}$ f... | Write down the unique prime factorization of $n = p_1^{k_1} \cdot p_2^{k_2} \cdot \ldots \cdot p_l^{k_l}$. Then $n \mid d^n$ implies $p_1 \cdot p_2 \cdot \ldots \cdot p_l \mid d$. Since $k_i < n$ for all $i \in \{1, \ldots , l\}$, we have $n \mid p_1^{n-1} \cdot p_2^{n-1} \cdot \ldots \cdot p_l^{n-1}$ and with $p_1^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/405913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\int_{-1}^{1}\frac{\arctan{x}}{1+x}\ln{\left(\frac{1+x^2}{2}\right)}dx$ This is a nice problem. I am trying to use nice methods to solve this integral, But I failed.
$$\int_{-1}^{1}\dfrac{\arctan{x}}{1+x}\ln{\left(\dfrac{1+x^2}{2}\right)}dx, $$
where $\arctan{x}=\tan^{-1}{x}$
mark: this integral is my favo... | FWIW, here's Maple:
> f:= arctan(x)/(1+x)*ln((1+x^2)/2);
> int(f, x=-1..1);
$$ {\frac {7}{64}}\,{\pi }^{3}-{\frac {5}{16}}\,\pi \, \left( \ln
\left( 2 \right) \right) ^{2}-\ln \left( 2 \right) {\it Catalan}+1/
2\, \left( \ln \left( 1-i \right) \right) ^{2}\pi -1/2\,i \left(
\ln \left( 1+i \right) \right) ^{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?
This should be easy, but I'm not seeing how. A hint would be appreciated.
| Putting everything together, suppose $x,y>0$. Then $0\le2\sqrt{xy}$. Hence:
$$
\sqrt{x+y} = \sqrt{x+0+y}
\le \sqrt{x+2\sqrt{xy}+y}
= \sqrt{(\sqrt{x}+\sqrt{y})^2}
= \sqrt{x}+\sqrt{y}
$$
as desired. Note that this relied on the fact that $f(x)=\sqrt{x}$ is monotonically increasing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/408177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 1
} |
Product of nilpotent matrices. Let $A$ and $B$ be $n \times n$ complex matrices and
let $[A,B] = AB - BA$.
Question: If $A , B$ and $[A,B]$ are all nilpotent matrices,
is it necessarily true that $\operatorname{trace}(AB) = 0$?
If,in fact, $[A,B] = 0$, then we can take $A$ and $B$ to be strictly upper triangular ... | The answer is no.
Take $A=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right)$, $B=XAX^{-1} = \left(\begin{array}{cccc} -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 1 & -1 & 1\end{array}\right)$, where we chose $X=\left(\begin{array}{cccc} 1 & 0 & 0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/408499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
Finding another way of doing this integral $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$ Problem :
Integrate : $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$
I have the solution : We can substitute $\sqrt{x}= \cos^2t$ and proceeding further,
I got the the answer which is $-2\sqrt{1-x}+\cos^{-1}\sqrt{x}+\sqrt{x-x^2}+C$
Can we do th... | You could also try $\sqrt{x}=tan\theta$.
$dx=2tan\theta sec^2\theta d\theta$
$\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}=\sqrt{\frac{1-tan\theta}{1+tan\theta}}$
Let $tan\theta=y$. Then $dy=sec^2\theta$. Substituting, we have
$\int{2y\sqrt{\frac{1-y}{1+y}}dy}=\int{\frac{2y(1-y)}{\sqrt{1-y^2}}dy}=\int{\frac{2y}{\sqrt{1-y^2}}d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Square roots of complex numbers I know that the square root of a number x, expressed as $\displaystyle\sqrt{x}$, is the number y such that $y^2$ equals x. But is there any simple way to calculate this with complex numbers? How?
| Write $(x+iy)^2=a+ib$, thus $x^2-y^2=a$ and $2xy=b$.
Now, $-x^2y^2=-b^2/4$, so $x^2$ and $-y^2$ are the solutions of $t^2-at-b^2/4=0$.
Then $\Delta=a^2+b^2$, and $t=\frac{a \pm \sqrt{a^2+b^2}}{2}$.
Obviously, since $x^2 \geq 0$ and $-y^2 \leq 0$, we have
$$x^2 = \frac{\sqrt{a^2+b^2}+a}{2}$$
$$y^2 = \frac{\sqrt{a^2+b^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/411174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
In how many ways can you split a string of length n such that every substring has length at least m? Suppose you have a string of length 7 (abcdefg) and you want to split this string in substrings of length at least 2.
The full enumation of the possibilities is the following:
ab/cd/efg
ab/cde/fg
abc/de/fg
abc/defg
abcd... | We can model the situation with generating functions. In order to do so we consider binary strings consisting of $0s$ and $1s$. Let $$0^\star=\{\varepsilon,0,00,000,\ldots\}$$ denote all strings containing only $0$s with length $\geq0$. The empty string is denoted with $\varepsilon$. The corresponding generating functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq i \leq n$? I need to prove this result for something I am working on:
For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ non... | Yes, your process can always be continued if you start with $m_1>n_1$ both odd.
$$
a_1 = m_1^2-n_1^2 \equiv 0 \pmod{4} \\
b_1 = 2m_1n_1 \equiv 2 \pmod{4} \\
c_1 = m_1^2+n_1^2 \equiv 2 \pmod{4}
$$
So $c_1/2$ is odd and we can factor it as $m_2\cdot n_2$ with $m_2>n_2$ both odd, take $m_2=c_1/2, n_2=1$ if necessary or an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How do I solve such logarithm I understand that
$\log_b n = x \iff b^x = n$
But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$)
What if I don't? For example, how do I solve for $x$ when:
$$\log_{1.03} 2 = x\quad ?$$
$$\log_{8} 33 = x\quad ?$$
| The logarithm $\log_{b} (x)$ can be computed from the logarithms of $x$ and $b$ with respect to a positive base $k$ using the following formula:
$$\log_{b} (x) = \frac{\log_{k} (x)}{\log_{k} (b)}.$$
So your examples can be solved in the following way with a calculator:
$$x = \log_{1.03} (2) = \frac{\log_{10} (2)}{\log_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Given that $x = 4\sin \left( {2y + 6} \right)$ find dy/dx in terms of x My attempt:
$\eqalign{
& x = 4\sin \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = \left( 2 \right)\left( 4 \right)\cos \left( {2y + 6} \right) \cr
& {{dx} \over {dy}} = 8\cos \left( {2y + 6} \right) \cr
& {{dy} \over {dx}} = {1 \ov... | To find $\frac{dy}{dx}$ in terms of $x$ for $x=4 \sin(2y+6)$, I wouldn't bother with implicit differentiation. Instead:
First, solve for $y$,
$$\frac{x}{4} = \sin(2y+6)$$
$$\arcsin(\frac{x}{4}) = 2y + 6$$
$$y = \frac{\arcsin(\frac{x}{4})-6}{2} = \frac{1}{2} \arcsin{\frac{x}{4}} - 3.$$
Now differentiate with respect to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
How to find the integral of $\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$ $$\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$$
What is the method to find an integral like this?
| As $\sin3x=3\sin x-4\sin^3x$
$$\sin^6\alpha=(\sin^3\alpha)^2=\left(\frac{3\sin\alpha-\sin3\alpha}4\right)^2$$
$$=\frac{9\sin^2\alpha+\sin^23\alpha-3(2\sin\alpha\sin2\alpha)}{16}$$
$$=\frac{9(1-\cos2\alpha)+1-\cos6\alpha-6(\cos\alpha-\cos3\alpha)}{32}$$
using $2\sin A\sin B=\cos(A-B)-\cos(A+B)$ and $\cos3x=1-2\sin^2x$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball
How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball?
My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ which is equ... | $$\sum_{r_1+r_2+r_3=5,r_i\geq1}\binom{5}{r_1}\binom{5-r_1}{r_2}\binom{5-r_1-r_2}{r_3}=$$
$$=\sum_{r_1+r_2+r_3=5,r_i\geq1}\frac{5!}{r_1!r_2!r_3!}=\frac{5!}{1!1!3!}+\frac{5!}{1!3!1!}+\frac{5!}{3!1!1!}+\frac{5!}{1!2!2!}+\frac{5!}{2!1!2!}+\frac{5!}{2!2!1!}=$$
$$=3(20+30)=150$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Find the volume using triple integrals Using triple integrals and Cartesian coordinates, find the volume of the solid bounded by
$$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$
and the coordinate planes $x=0, y=0,z=0$
My take
I have set the parameters to $$ 0\le x \le a$$ $$0\le y \le b\left( 1 - \frac{x}{a} \right)$$ ... | Here is an alternative computation using a single variable integral that confirms your result. The following figure represents the given pyramid.
The equations of the lines situated on the planes $y=0$ and $z=0$ are:
$$y=0,\qquad\frac{x}{a}+\frac{z}{c}=1\Leftrightarrow z=\left( 1-\frac{x}{a}\right) c,$$
$$z=0,\qquad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$ How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$
This is about as far as I got:
$$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$
$$\sum_{j=2}^\infty\frac{2^j j!}{(j)_j} = \frac{2^2 2!}{... | This is going to be a little out of the blue, but here goes.
Consider the function
$$f(x) = \frac{\arcsin{x}}{\sqrt{1-x^2}}$$
$f(x)$ has a Maclurin expansion as follows:
$$f(x) = \sum_{n=0}^{\infty} \frac{2^{2 n}}{\displaystyle (2 n+1) \binom{2 n}{n}} x^{2 n+1}$$
Differentiating, we get
$$f'(x) = \frac{x \, \arcsin{x}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 4,
"answer_id": 1
} |
Prove that $\log _5 7 < \sqrt 2.$
Prove that $\log _5 7 < \sqrt 2.$
Trial : Here $\log _5 7 < \sqrt 2 \implies 5^\sqrt 2 <7.$ But I don't know how to prove this. Please help.
| Observe that:
$$ \begin{align*}
\log_5 7 &= \dfrac{3}{3}\log_5 7 \\
&= \dfrac{1}{3}\log_5 7^3 \\
&= \dfrac{1}{3}\log_5 343 \\
&< \dfrac{1}{3}\log_5 625\\
&= \dfrac{1}{3}\log_5 5^4\\
&= \dfrac{1}{3}(4)\\
&= \sqrt{\dfrac{16}{9}}\\
&< \sqrt{\dfrac{18}{9}}\\
&= \sqrt{2}\\
\end{align*} $$
as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/423348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
} |
The dimension of the vector space of all trace-zero symmetric matrices Find the dimension of the vector space of all symmetric matrices of order $n \times n$ (real entries) and trace equal to zero.
| $$V_F=\{A\in M_M(F)\:A^T=A , Trace(A=0)\}=$$ forall $A\in V_F$ we have $trac(A)=0$ and $a_{ij}=a_{ji}$ such that $A=[a_{ij}]_{n \times n}$
$$A=\begin{pmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
a_{12} & a_{22} & a_2^2 & \cdots & a_2^n \\
\vdots & \vdots& \vdots & \ddots & \vdots \\
a_{1n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving that a graph is self complementary I've been given the following adjacency matrix:
$$\left(\begin{array}{cccccccc} 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 &
1 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 &
1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 &
1... | The blue graph has the big square 2-4-6-8 with little triangles 2-3-4, 4-5,6, 6-7-8, and 8-1-2, and also the little triangle vertices which are opposite relative to the big square are joined, i.e. 1-5 and 3-7.
The red graph has a similar description: there is a big square 1-3-5-7, and triangles 1-6-3, 3-8-5, 5-2-7, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximum of a trigonometric function without derivatives I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or identity?
$f(x)=\sin(2x)+2\sin(x)$
| The idea is to use $\sin^2 x + \cos^2x = 1$ to reduce to dealing with only 1 trigonometric function, and then proceed as a standard 1-variable inequality.
We wish to find the maximum of $f(x) = \sin 2x + 2 \sin x = 2 \sin x ( 1 + \cos x )$.
It is clear that we may assume $\sin x \geq 0, \cos x \geq 0$, to maximize this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Divisibility by Quadratics $b^2+ba+1\mid a^2+ab+1\Rightarrow\ a=b$
The natural numbers $a$ and $b$ are such $a^2+ab+1$ is divisible by $b^2+ba+1$. Prove that $a = b$.
I tried to algebraically manipulate it as follows:
$(b^2 + ba + 1)k = a^2 + ab + 1$
$[b(a + b) + 1]k = a(a + b) + 1$
$kb(a + b) + k = a(a + b) + 1$
$k ... | If $a^2+ab+1$ is divisible by $b^2+ba+1$, then so is $(a^2+ab+1)-(b^2+ba+1)=a^2-b^2$.
Note that $a+b$ and $b^2+ba+1$ are relatively prime. So $b^2+ab+1$ divides $a-b$. Now you should be able to finish, using considerations of size.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/429638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Factor $x^4 - 11x^2y^2 + y^4$ This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer.
The question is:
Factor $x^4 - 11x^2y^2 + y^4$
The answer is:
$(x^2 - 3xy -y^2)(x^2 + 3xy - y^2)$
My question is:
How did the textbook get this?
I tried the following methods (exa... | Note that the answer is not unique. There are four linear factors, and you can get different quadratic factors by grouping them the other way.
For instance:
$$
\begin{align*}
x^4 - 11x^2y^2 + y^4
&= x^4 + 2x^2y^2 + y^4 - 13x^2y^2 \\
&= (x^2+y^2)^2-(\sqrt{13}xy)^2 \\
&= (x^2+y^2-\sqrt{13}xy)(x^2+y^2+\sqrt{13}xy).
\end{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 1
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Limit on the expression containing sides of a triangle To find the bounds of the expression $\frac{(a+b+c)^2}{ab+bc+ca}$, when a ,b, c are the sides of the triangle.
I could disintegrate the given expression as $$\dfrac{a^2+b^2+c^2}{ab+bc+ca} + 2$$ and in case of equilateral triangle, the limit is 3.
Now how to procee... | Without loss of generality, $a \leqslant b \leqslant c = 1$. Thus it remains to see that
$$1 \leqslant \frac{1+a^2+b^2}{a + b + ab} \leqslant 2.$$
For the left inequality, we rewrite
$$a + b + ab \leqslant 1 + a^2 + b^2 \iff 0 \leqslant 1 - a - b + ab + a^2 -2ab + b^2 = (1-a)\cdot(1-b) + (b-a)^2,$$
which is evident.
Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Covergence of $\frac{1}{4} + \frac{1\cdot 9}{4 \cdot 16} + \frac{1\cdot9\cdot25}{4\cdot16\cdot36} + \dotsb$ I am investigating the convergence of the following series: $$\frac{1}{4} + \frac{1\cdot 9}{4 \cdot 16} + \frac{1\cdot9\cdot25}{4\cdot16\cdot36} + \frac{1\cdot9\cdot25\cdot36}{4\cdot16\cdot36\cdot64} + \dotsb$$
T... | There are several methods as commented already, but I will put mine for what is worth.
Here's my answer:
Let
$$
a_n=\frac 1 2 \cdot \frac 3 4 \cdots \frac{2n-1}{2n}$$
Then
$$
a_n^2=\frac 1 2 \frac 1 2 \frac 3 4 \frac 3 4 \cdots \frac{2n-1}{2n}\frac{2n-1}{2n}$$
$$a_n^2 > \frac 1 2 \frac 1 2 \frac 2 3 \frac 3 4 \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
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Finding the relation between function x,y,z - trigo problem Problem :
For $\displaystyle 0 < \theta < \frac{\pi}{2}$ if
$$\begin{align}x &= \sum^{\infty}_{n =0} \cos^{2n}\theta \\ y &= \sum^{\infty}_{n =0} \sin^{2n}\theta\\ z &= \sum^{\infty}_{n =0} \cos^{2n}\theta \sin^{2n}\theta \end{align}$$
then
options are :
(... | $$\begin{align}x &= \sum^{\infty}_{n =0} \cos^{2n}\theta \\ y &= \sum^{\infty}_{n =0} \sin^{2n}\theta\\ z &= \sum^{\infty}_{n =0} \cos^{2n}\theta \sin^{2n}\theta \end{align}$$
$$x=1+{\cos^2\theta}+{\cos^4\theta}+\cdots\infty \;terms$$
this is infinite geometric series :
$S_{\infty}=\dfrac {a}{1-r}\;\;\,|r|<1$ ,here $\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How many times does the function $y=e^x $ meet $y=x^2$? As you know $y=e^x$ and $y= x^2$ meet once on $x<0$.
But I want to know whether or not they meet on $x>0$.
Since $\lim_{x\rightarrow \infty } e^x/x^2=\infty$, if they meet once on $x>0$, they
must meet again.
To summarize, my question is whether or not they m... | Note that $$e^x - x^2 = \left(e^{x/2}-x\right)\left(e^{x/2}+x\right)$$
For $x\ge 0$, the second factor is strictly positive, so consider the first factor:
$$\begin{align}e^{x/2}-x &= \left(1+\frac{x}{2}+\frac{x^2}{8} + \frac{x^3}{48} + \cdots\right) - x \\
&= 1 - \frac{x}{2}+\frac{x^2}{8} + \frac{x^3}{48}+\cdots \\[6pt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$ $$\int \frac{dx}{\sqrt{x^2 -9}}$$
$x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$
$$\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta \tan\theta d\theta}{\tan\theta} \\ \\ & = \int \sec\theta d\theta \\ \\ & = \ln | \sec\the... | What you should end with is
$\sec\theta = \dfrac x3\quad$ and $\quad\tan \theta = \dfrac{\sqrt{ x^2 - 9}}{3}$.
Then you have $$\begin{align} \log \Big| \frac 13\left(x + \sqrt{x^2 - 9}\right)\Big| + C & = \log|x +\sqrt{x^2 - 9}| -\log 3 + C \\ \\ & = \log|x + \sqrt{x^2 - 9}| + C'\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/436153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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what is the explicit form of this iterativ formular I am not sure, if there is an explicit form, but if there is, how do I get it?
This is the formula:
$$c_n=\frac{1-n \cdot c_{n-1}}{\lambda}$$
where $\lambda \in \mathbb{R}$ and $n \in \mathbb{N}$
I already tried some forms for c via trail and error, but I couldn't fin... | We have $$\begin{align} c_{10} & = \tfrac 1\lambda - \tfrac{10}\lambda c_9 \\
& = \tfrac 1\lambda - \tfrac{10}{\lambda^2} (1-9c_8) \\
& = \tfrac 1\lambda - \tfrac{10}{\lambda^2} + \tfrac{10\cdot 9}{\lambda^2} c_8 \\
& = \tfrac 1\lambda - \tfrac{10}{\lambda^2} + \tfrac{10\cdot 9}{\lambda^3} -\tfrac{10\cdot 9\cdot 8}{\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Volume using disk method $y = 2\sqrt{x} \quad y=x$ $y = 2\sqrt{x}$
$y=x$
about $x=-2$
I know that $y=x$ is on the outside and they meet at 4.
I need these in terms of y since I rotate about y.
$x = y$
$x = \frac{y^2}{4}$
$$\pi \int_0^4 (y - (-2))^2 - \left(\frac{y^2}{4} - (-2)\right)^2 dy$$
$$\pi \int_0^4 (y + 2)^2 - \... | The issue appears to be in your second expansion. $$\begin{align}\left(\frac{y^2}4-(-2)\right)^2 &= \left(\frac{y^2}4+2\right)^2\\ &= \left(\frac{y^2}4\right)^2+2\left(\frac{y^2}4\right)(2)+(2)^2\\ &= \frac{y^4}{16}+y^2+4.\end{align}$$
Can you get the rest of the way from there?
| {
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"url": "https://math.stackexchange.com/questions/438271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that, if $0 < x < 1$, then $(1+\frac{x}{n})^n < \frac1{1-x}$ More fully,
if $n\ge 2$ is an integer
and
$0 < x < 1$,
prove that
$(1+\frac{x}{n})^n < \frac1{1-x}$.
In addition,
if $c > 1$ and
$0 < x \le \frac{c-1}{c}$,
prove that
$(1+\frac{x}{n})^n < 1+cx$.
Proofs by elementary means
(no calculus or limits)
are pa... | For the first part, we only need the binomial theorem:
$$\left(1 + \frac{x}{n}\right)^n = \sum_{k=0}^n \binom{n}{k}\frac{x^k}{n^k} = \sum_{k=0}^n \frac{\prod_{j=1}^k(n+1-j)}{k!n^k}x^k \leqslant \sum_{k=0}^n \frac{x^k}{k!} \leqslant \sum_{k=0}^n x^k < \frac{1}{1-x}.$$
For the second, we observe that
$$\frac{1}{1-x} \leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots.
I am able to prove that all roots would be negative .How to proceed ... | This is a symmetric (the coefficients) polynomial. We may want to take advantage of that. We can divide by $x^2$ and get $$(x+\frac{1}{x})^2+(a-1)(x+\frac{1}{x})-1=0$$
From there we can solve for $x+\frac{1}{x}$ and get $$x+\frac{1}{x}=\frac{-(a-1)\pm\sqrt{(a-1)^2+4}}{2}.$$
This is a quadratic equation. Solve for $x$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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An identity which applies to all of the natural numbers Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.
| I like this approach to the solution of this equation.
If we consider the Diophantine equation: $qX^2+Y^2=Z^2+j$
If the root is a : $a=\sqrt{\frac{j}{q}}$
We use the solutions of Pell's equation: $p^2-(q+1)s^2=1$
Solutions can be written:
$X=2s(s\pm{p})L\pm{ap^2}+2aps\pm{a(q+1)s^2}=bL+af$
$Y=(p^2\pm2ps+(1-q)s^2)L\... | {
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Find $ \int \frac {\tan 2x} {\sqrt {\cos^6x +\sin^6x}} dx $
Problem: Find $\displaystyle\int \frac {\tan 2x} {\sqrt {\cos^6 x +\sin^6 x}} dx $
Solution: $\tan 2x= \dfrac{2\tan x}{1-\tan^2 x}$
Also I can take $\cos^6x$ common from $\sqrt {\cos^6x +\sin^6x}$
I don't know whether it is good approach to the question
Ple... | HINT:
$$\cos^6x+\sin^6x=(\cos^2x+\sin^2x)^3-3\cos^2x\sin^2x(\cos^2x+\sin^2x)$$
$$=1-3\cos^2x\sin^2x=1-\frac34(\sin2x)^2$$
$$=1-3\cos^2x\sin^2x=1-
\frac34(\sin2x)^2=1-\frac34(1-\cos^22x)=\frac{1+3\cos^22x}4$$
Use $\cos2x=u$
| {
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If $x,y,z \in \Bbb{R}$ such that $x+y+z=4$ and $x^2+y^2+z^2=6$, then show that $x,y,z \in [2/3,2]$. If $x,y,z\in\mathbb{R}$ such that $$x+y+z=4,\quad x^2+y^2+z^2=6;$$then show that the each of $x,y,z$ lie in the closed interval $[2/3,2]$.
I have been able to solve using $2(y^2+z^2)\geq(y+z)^2$.
Is there any another met... | The mean of the numbers is $\frac{4}{3}$, while the mean of their squares is $2$,
that is
$$E(x)= \bar x =\frac{1}{n}\sum x= \frac{4}{3} \\
E(x^2) = \frac{1}{n}\sum x^2 = 2$$
($n$ is $3$)
Therefore, the variance is
$$\frac{1}{n}\sum(x - \bar x)^2 = E(x^2) - E(x)^2 = 2 - (\frac{4}{3})^2 = \frac{2}{9}$$
In other words... | {
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Need help using the alternative formula to find the derivative at $x = c$ Don't know if I'm simplifying wrong or if it is a simple mistake but I cannot get the answer of 4. Please help and include exact step by step details. Thank you.
$$f(x)=x^3 + 2x^2 + 1,\;\;\;c= -2$$
| Using the alternative definition of the derivative, given what you posted in a comment, we'll start with the approximation of the derivative given by $\lim_{x \to -2}\dfrac{f(x) - f(c)}{x - c}$
\begin{align*}\require{cancel}
\lim_{x \to -2} \dfrac{f(x) - f(c)}{x - c}
&=\lim_{x\to -2} \dfrac{x^3 + 2x^2 + 1 - ((-2)^3 + 2... | {
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using Gauss' algorithm (for linear congruences) for A > B To solve $Bx \equiv A \pmod{m}$, use Gauss' algorithm.
The algorithm works perfectly when $A < B$. For example, to solve $6x \equiv 5 \pmod{11}$: $$x \equiv \frac{5}{6} \equiv \frac{5(2)}{6(2)} \equiv \frac{10}{12} \equiv \frac{10}{1}$$
so $x \equiv 10$
But when... | Frank, you can use Gauss's Algorithm even if modulo is not prime. The only thing you need to take care is that multiplier should be co-prime to modulo.
Just keep multiplying denominator by a number so that denominator is near 100 till denominator become 1. However, the multiplier must be co-prime to 100.
$$\frac{13}{7}... | {
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Properties of Triangle - Trigo Problem : In $\triangle $ABC prove that $a\cos(C+\theta) +c\cos(A-\theta) = b\cos\theta$ Problem :
In $\triangle $ABC prove that $a\cos(C+\theta) +\cos(A-\theta) = b\cos\theta$
My approach :
Using $\cos(A+B) =\cos A\cos B -\sin A\sin B and \cos(A-B) = \cos A\cos B +\sin A\sin B$, we get... | Which means that $c \sin(B+C)=a \sin(A+B)$
$$c\sin(B+C)-a\sin(A+B)=O$$
| {
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"url": "https://math.stackexchange.com/questions/449596",
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"question_score": "3",
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"Question: Show that $n^5 - n$ is divisible by 30; for all natural n"
Show that $n^5 - n$ is divisible by $30;$ $\forall n\in \mathbb{N}$
I tried to solve this three-way. And all stopped at some point.
I) By induction:
testing for $0$, $1$ and $2$ It is clearly true.
As a hypothesis, we have $30|n^5-n\Rightarrow n^5-... | Method $1:$
$$n^5-n=n(n^4-1)=n(n^2-1)(n^2+1)=n(n^2-1)(n^2-4+5)$$
$$=n(n^2-1)(n^2-4)+5n(n^2-1)$$
$$=\underbrace{(n-2)(n-1)n(n+1)(n+2)}_{\text{ product of }5\text{ consecutive integers }}+5\underbrace{(n-1)n(n+1)}_{\text{ product of }3\text{ consecutive integers }}$$
Now, we know the product $r$ consecutive integers is... | {
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Maclaurin series of $f(x) = e^x \sin x$ $$f(x) = e^x \sin x$$
I tried applying the given formula in my book but it didn't work.
The maclaurin for $e^x$ is given as $\displaystyle \sum \frac{x^n}{n!}$ and $\sin x$ $\displaystyle \sum \frac{(-1)^n x^{2n + 1}}{(2n+1)!}$
I attempted to multiply them together, failed teh bo... | Since you are looking for the Maclaurin Series of $ f(x):=\exp(x)\sin(x) $ you can start by looking at $f(0)$ and higher derivatives of $f$ at $x=0$. Let $f^{(n)}$ denote the $n$-th derivative of $f$:
$$
\begin{alignat}{2}
f(0) &= \exp(0)\sin(0) &&{}= 0 \\
f^{(1)}(0) &= \exp(0)\sin(0)+ \exp(0)\cos(0) &&{}= 1 \\
f^{(2)}... | {
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How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$? How to find the sum of the following series?
$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$$
This is a harmonic progression. So, is the following formula correct?
$\frac{(number ~of ~terms)^2}{sum~ of~ al... | The exact expression for $\displaystyle H_n:=1+\frac{1}{2}+\frac{1}{3}+\cdots\ +\frac{1}{n} $ is not known, but you can estimate $H_n$ as below
Let us consider the area under the curve $\displaystyle \frac{1}{x}$ when $x$ varies from $1$ to $n$.
Now note that $\displaystyle H_{n}-\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\... | {
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Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$
Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$
I know that $\sin2\alpha = 2\sin\alpha\cos\alpha$
so
$$\sin2\alpha\cos\alpha=2\sin\alpha\cos^2\alpha$$
and $\... | Hint
Take the real part of this complex number
$$-ie^{i\alpha}e^{2i\alpha}=-ie^{-i\alpha}e^{4i\alpha}$$
| {
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"source": "stackexchange",
"question_score": "2",
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Condition for collinearity of points $(a, a^3), (b, b^3), and (c, c^3)$ The following is a statement I have been trying to prove (while solving problem 1.4.26 in Algorithms (4th edition) by Robert Sedgewick).
Show that three points $(a, a^3), (b, b^3), and
(c, c^3)$ are collinear if and only if $a + b + c = 0$.
I ... | We need $$\det\begin{pmatrix} a^3 & a & 1 \\ b^3 & b & 1 \\ c^3 & c &1\end{pmatrix}=0$$
Now,
$$\det\begin{pmatrix} a^3 & a & 1 \\ b^3 & b & 1 \\ c^3 & c &1\end{pmatrix}$$
$$=\det\begin{pmatrix} a^3-b^3 & a-b & 0 \\ b^3 & b & 1 \\ c^3-b^3 & c-b &0\end{pmatrix}$$
$$=-(a-b)(b-c)\det\begin{pmatrix} a^2+ab+b^2 & 1 & 0 \\ b^... | {
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"source": "stackexchange",
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$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
| $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc=a^{2}+b^{2}+c^{2}+2abc\big(\frac{1}{c}+\frac{1}{b}+\frac{1}{a}\big)=a^{2}+b^{2}+c^{2}$
since $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$.
| {
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Checking Independence for $\left\{\dfrac{1}{x^2+x-6},\dfrac{1}{x^2-5x+6},\dfrac{1}{x^2-9}\right\}$?
Is this subset of $V=\mathbb{F[0,1]}$ linearly independent, where the subset of $V$ is
$\left\{\dfrac{1}{x^2+x-6},\dfrac{1}{x^2-5x+6},\dfrac{1}{x^2-9}\right\}$?
My work:
If this subset is independent, then $\alpha=... | Hint: checking that this set is linearly independent is equivalent to checking if the set
$$\{x-3, x-2, x+3\}$$ is linearly independent.
| {
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"source": "stackexchange",
"question_score": "2",
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Diophantine equation $x^3=a^2+b^2+c^2$ Does anyone know if a formula exists to obtain all solutions of the above Diophantine equation? All numbers integers.
Addendum: After seeing the answer from Tito Piezas III, I reconsidered the above equation and came up with an original solution that applies to two consecutive cu... | For any positive integer $k>2$, the kth power of the sum of two squares is the sum of three squares,
$$x^2+y^2+z^2 = (a^2+b^2)^k$$
thus,
$$a^2(a^2+b^2)^2 + (2ab^2)^2 + (a^2-b^2)^2b^2 = (a^2+b^2)^3$$
$$(a^4-b^4)^2 + (4a^2b^2)^2 + (2ab(a^2-b^2))^2 = (a^2+b^2)^4$$
$$a^2(a^2+b^2)^4 + (4a^3b^2-4ab^4)^2 + (a^4b-6a^2b^3+b^5)... | {
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How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$
How to compute
$$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$
My Working :
$$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1}
= 1-0 = 1$$
Is it correct
| Note that
$${n^3 - 1 \over n^3 + 1} = {n - 1 \over n + 1}{n^2 + n + 1 \over n^2 - n + 1}$$
Also note that
$$(n-1)^2 + (n - 1) + 1 = n^2 - n + 1$$
So the infinite product in question is really the product of two telescoping products.
| {
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"source": "stackexchange",
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"answer_id": 1
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Differentiation of logarithmic functions using the chain rule What's the derivative of $x^2(\ln(x^2))$?
I'm having a really hard time with logarithmic differentiation. Can someone help rationalize it for me?
| Hints:
*
*$\bigl( f \cdot g \bigr)' = f \cdot g' + g \cdot f'$.
Using this we see that if we see that
\begin{align*}y &= x^{2} \cdot \log(x^{2})\\
\implies y' &= x^{2} \cdot \frac{d}{dx}(\log(x^{2})) + \log(x^{2}) \cdot \frac{d}{dx} (2x) \\ &= x^{2} \cdot \frac{d}{dx} (2\cdot \log(x)) + \log(x^{2}) \times 2 \\
&=... | {
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Proving an equality involving binomial coefficients and summations Question:
$$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right )^{k}$$
Attempt:
It looks like I need to start with a quadr... | Suppose we seek to verify that
$$\sum_{k=0}^n (-1)^k {2n\choose k} {2n-k\choose 2n-2k}
= \sum_{k=0}^{2n} {2n\choose k}^2
\left(\frac{1+\sqrt{5}}{2}\right)^{2n-k}
\left(\frac{1-\sqrt{5}}{2}\right)^{k}.$$
For the LHS introduce
$${2n-k\choose 2n-2k} =
\frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{1}{z^{2n-2k+1}} (1+z)^{2n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463521",
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"source": "stackexchange",
"question_score": "3",
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Identifying the nature of the eigenvalues I wish somebody could help me in this one. We have to choose one of the $4$ options.
Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. Consider the $3 \times 3$ matrix
$$A=\begin{bmatrix}
1 & b & c \\
b & a & 0 \\
c & 0 & 1 \\
\end{bm... | Notice that the matrix is symmetric (so, we can exclude option 4 immediately), and the question basically asks if the matrix is negative definite (all eigenvalues are negative), positive definite (all eigenvalues are positive), or indefinite (we have both negative and positive eigenvalues).
Let's check the leading prin... | {
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Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$ The following is an absolute value quadratic equation that I want to solve:
$$x^2 − 6 |x − 2| − 28 = 0$$
Here is what I did :
$x^2 − 6 |x − 2| − 28 = 0$
$x^2 − 6 |x − 2| − 28 = 0$
$-6|x-2|=28-x^2$
$6|x-2|=x^2-28$
$6x-12=x^2-28$ or $28-x^2$
(Is this step correct ?)
Sol... | HINT:
First of all, let $x=a+ib$ where $a,b$ are real
So, we have $$(a+ib)^2-28=6|a+ib-2|\implies a^2-b^2-28+2ab i=6\sqrt{(a-2)^2+b^2}$$
Equating the imaginary parts, we have $ab=0$
If $a=0,-(b^2+28)=6\sqrt{4+b^2}>0$ which is impossible
So, $b$ must be $0\implies x$ must be real
We know for real $m,$ $$|m|=\begin{cases... | {
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Double Integral Question on unit square Hints on solving following double integral will be appreciated.
$$\int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x$$
| Just split the integral as
$$ \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x
= \int_0^1 \int_0^1 \frac{x^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x
- \int_0^1 \int_0^1 \frac{y^2}{(x^2+y^2)^2} \text{d}y \ \text{d}x. $$
Now, just use standard integration techniques to evaluate the integrals. For instance... | {
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equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$ In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds:
$3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$
Please prove this identi... | A Cayley-Menger determinant gives the volume of "tetrahedron" $PABC$:
$$288\; V^2 = \left|\begin{array}{ccccc}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & d^2 & d^2 & a^2 \\
1 & d^2 & 0 & d^2 & b^2 \\
1 & d^2 & d^2 & 0 & c^2 \\
1 & a^2 & b^2 & c^2 & 0
\end{array}\right| = d^2 \left( \left( a^2+b^2+c^2+d^2 \right)^2-3\left(a^4+b^4+c^... | {
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proof for the Ramanujan's formula ? I found this formula in a textbook in which the proof to the formula was not given
Ramanujam's formula
$$\sqrt{1 +n\sqrt{1 +(n+1)\sqrt{1 + (n+2)\sqrt{1 + (n+3)\sqrt{1 +....\infty}}}}} = n+1$$
Its a great equation andhow do you prove this. its a bit difficult for me and tried differen... | More general formulation can be gotten:
$F(x) = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{a(x+2n)+(n+a)^2+(x+2n)\sqrt{\mathrm{\cdots}}}}}$
$$F(x)^2 = ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\mathrm{\cdots}}} $$
Which can be simplified to:
$$F(x)^2 = ax+(n+a)^2 +xF(x+n) \tag1$$
Let's assume that $F(x) = ... | {
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Prove that $\lceil(\sqrt{3}+1)^{2n}\rceil$ is divisible by $2^{n+1}$. Let $n$ be a positive integer. Prove that $\lceil(\sqrt{3}+1)^{2n}\rceil$ is divisible by $2^{n+1}$.
I tried rewriting $\lceil(\sqrt{3}+1)^{2n}\rceil$ as $m*2^{n+1}$ for some m, but couldn't get anywhere.
| Hint: The value is equal to $( \sqrt{3} + 1)^{2n} + (\sqrt{3}-1 )^{2n}$
Hint: Expanding the square, we get $ (4 + 2\sqrt{3})^n + (4-2\sqrt{3})^n$
This term is clearly a multiple of $2^n$, which we can factor out.
Hint: $(2 + \sqrt{3})^n + (2-\sqrt{3})^n$ is even, by the Binomial Theorem expansion.
| {
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Evaluate $\int \limits _{0}^{\infty}\ln\left({x+\frac{1}{x}}\right)\cdot\frac{\mathrm dx}{1+x^2}$ $$\int \limits _{0}^{\infty}\ln\left({x+\frac{1}{x}}\right)\cdot\frac{dx}{1+x^2}$$
we are asked to solve this definite integral so here's what i did
$$\int \limits_{0}^{\infty}\ln \left({\frac{x^2 +1}{x}}\right)\cdot\frac{... | As you have done the integral can be fairly easy evaluated by splitting it into two easier integrals.
$$
\int_{0}^{\infty}\log \left({\frac{x^2 +1}{x}}\right)\cdot\frac{dx}{1+x^2}\mathrm{d}x =
\int_{0}^{\infty}\frac{\log\left(x^2+1\right)}{1+x^2}\,\mathrm{d}x -
\int_{0}^{\infty}\frac{\log x}{1+x^2}\,\math... | {
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How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly? I found this amazingly beautiful identity here. How to prove that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ without directly multiplying the factors? (I've already verified it that way). ... | We know $(A+B)\mid(A^n+B^n)$ for $n$ odd. What about with three terms? Compute
$$\mod A+B+C:\quad A^3+B^3+C^3\equiv-(B+C)^3+B^3+C^3\equiv-3BC(B+C)\equiv 3ABC.$$
So $(A+B+C)\mid(A^3+B^3+C^3-3ABC)=f(A,B,C)$. Further
$$f(A,B,C)=f(A,\omega B,\bar{\omega}C)=f(A,\bar{\omega}B,\omega C)=\rm etc.$$
by inspection so both $A+\om... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 6,
"answer_id": 5
} |
Summation of series Write down the sum of $\displaystyle \sum_1^{2N} n^3$ in terms of $N$, and hence find:
$1^3 - 2^3 + 3^3 - 4^3 + \cdots - (2N)^3$ in terms of $N$, simplifying your answer.
I found this to be $n^2(2n+1)^2$ but the next part is not making sense to me.
Why is the general term of this sum $-(2N)^3$, wher... | HINT:
As
$$
\begin{align}
& \sum_{1\le r\le n}r^3=\frac{n^2(n+1)^2}4 \\[10pt]
& 1^3 - 2^3 + 3^3 - 4^3 + \cdots - (2N)^3 \\[10pt]
& =\sum_{1\le r\le 2N}r^3-2\sum_{1\le r\le N}(2r)^3 \\[10pt]
& =\sum_{1\le r\le 2N}r^3-2\cdot8\sum_{1\le r\le N}r^3 \\[10pt]
& =\frac{(2N)^2(2N+1)^2}4-16\frac{N^2(N+1)^2}4 \\[10pt]
& =N^2\{(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/475799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find all $x,y,z \in \mathbb{Q}.x^2+y^2+z^2+x+y+z=1$ Find all $x,y,z \in \mathbb{Q}$
where.$x^2+y^2+z^2+x+y+z=1$
| Assume $(x,y,z)$ is sastifying the equation.
We can multiply $K=N^2(N \in \mathbb{Z}^+)$ to the both sides of equation.
And the both sides are integers.
Set $N=2^{h}s$,$s$ is odd.
Then $K((2x+1)^2+(2y+1)^2+(2z+1)^2)=7N^2$
The $LHS$ is sum of 3 squares.
From A production by Gauss, $a^2+b^2+c^2=m,(a,b,c,m,\alpha,t \in\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding minima and maxima of $\sin^2x \cos^2x$ I am trying to find the turning points of $\sin^2x \cos^2x$ in the range $0<x<\pi$.
So far I have:
$\begin{align} f'(x) & =\cos^2x \cdot 2\sin x \cdot \cos x + \sin^2x \cdot 2\cos x \cdot -\sin x \\
& = 2 \cos^3x \sin x - 2\sin^3x \cos x \\
& = 2 \sin x \cos x (\cos^2x - \... | The equation $\cos x+\sin x=0$ is really the same as $\cos x-\sin x=0$ once you make the transformation $x\mapsto x+\frac{\pi}2$, because $\cos (x+\frac{\pi}2)=-\sin x$ and $\sin (x+\frac{\pi}2)=\cos(x)$. The latter can be solved by resorting to $\cos ^2+\sin ^2=1$.
Edit: We can derive $\cos (x+\frac{\pi}2)=-\sin x$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/477636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Prove that:$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$ Find $n\in\mathbb{N}^+$
For all Positive real numbers $a,b,c$ sastifying $a+b+c=3$
$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$
| For an upper bound, consider $(a,b,c)=\left(\frac{3}{2},\frac{3}{4},\frac{3}{4}\right)$. (I thought of this myself, but I'm going to make it Community Wiki as this is reflected in the AoPS page.)
We need to solve for the smallest $n$ such that:
$\left(\frac{2}{3}\right)^n+2\left(\frac{4}{3}\right)^n<\left(\frac{3}{2}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/479074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
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Fractions in Questions and Answers
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