qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,098,350 | <p>Recently I was able to find a result to a common definite integral:
<span class="math-container">\begin{equation}
J(n_1, k_1, m_1) = \int_0^{\infty} \frac{x^{k_1}}{\left(x^{n_1} + a_1 \right)^{m_1}}\:dx = \frac{a^{\frac{k_1 + 1}{n_1} - m_1}}{n_1}\Gamma\left(m_1 - \frac{k_1 + 1}{n_1}\right)\Gamma\left(\frac{k_1 + 1}... | Yuri Negometyanov | 297,350 | <p>HINT</p>
<p>Let us start from the OP result
<span class="math-container">\begin{align}
&H_1\left(a_1, a_2, m_1, m_2, n_3, k_3\right)
= \frac{1}{n_2}\int\limits_0^{\infty} \frac{u^{k_3}}{\left(u^{n_3} + a_1\right)^{m_1}} \frac{1}{\left(u + a_2\right)^{m_2}}\:du.\tag1
\end{align}</span></p>
<p>Taking in account... |
3,098,350 | <p>Recently I was able to find a result to a common definite integral:
<span class="math-container">\begin{equation}
J(n_1, k_1, m_1) = \int_0^{\infty} \frac{x^{k_1}}{\left(x^{n_1} + a_1 \right)^{m_1}}\:dx = \frac{a^{\frac{k_1 + 1}{n_1} - m_1}}{n_1}\Gamma\left(m_1 - \frac{k_1 + 1}{n_1}\right)\Gamma\left(\frac{k_1 + 1}... | Maxim | 491,644 | <p>This is an integral of two Meijer G-functions which gives a Fox H-function. Let <span class="math-container">$\nu_1 = 1/n_1$</span>, <span class="math-container">$\,\nu_2 = 1/n_2$</span>, <span class="math-container">$\,\sigma = k_1 + k_2 + 1$</span>. The Mellin transform is
<span class="math-container">$$\mathcal M... |
1,025,088 | <p>I have been working on a problem as follows:
Do there exist 100 consecutive natural numbers none of which is prime?
I know that the answer is 'yes', by considering 101!, and noting the sequence 101! + 2, 101! + 3, ... , 101! + 101.</p>
<p>This approach generalises nicely by considering (n+1)!</p>
<p>However, whils... | Asvin | 68,188 | <p>One way to show this is by the <a href="http://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" rel="nofollow">divergence of sum of reciprocals of primes</a>. If the density of primes was a constant $k$, then the n-th prime $p_n \geq kn$. This will show that the sum diverges. This does n... |
226,353 | <p>I want to solve the following equation</p>
<pre><code>y''[x] + a + b y[x] + c y[x]^2 == 0, y[∞] == 0, y'[∞] == 0
</code></pre>
<p>where <code>a</code>, <code>b</code> and <code>c</code> are constants.</p>
| Artes | 184 | <p><code>DSolve</code>yields an <strong>involved implicit result</strong> in terms of elliptic integrals. In order to solve explicitly the problem at hand we should first transform the equation. Let's multiply given differential equation by <span class="math-container">$\;y'(x)$</span>
<span class="math-container">$$y'... |
4,343,792 | <p>Suppose that <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> are both prime numbers.</p>
<p><strong>a) Is the integer between <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> odd or even? Explain your answer.</strong></p>
<p>All prime numbers... | poetasis | 546,655 | <p>Given<span class="math-container">$\quad A^2+B^2=C^2\quad $</span> the most common means of generating Pythagorean triples is Euclid's formula
<span class="math-container">$A=m^2-k^2,\quad B=2mk, \quad C=m^2+k^2.\quad$</span> We can restrict this formula to generating only primitive triples and odd square multiples ... |
2,012,371 | <p>I am new to calculus. Can you help me with this?</p>
<h2>$a+\sqrt{a}=4$</h2>
<p>$5a+a\sqrt{a}=? $</p>
| Lutz Lehmann | 115,115 | <p>Set $a=x^2$, $x=\sqrt a$ to obtain basic quadratic equations.</p>
|
175,791 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/173370/compute-the-trigonometric-integrals">Compute the trigonometric integrals</a> </p>
</blockquote>
<p>For $n \in \mathbb N$, $$ \int_{-\pi}^{\pi} \frac{1 - \cos (n+1) x}{1- \cos x} dx = (n+1) 2 \pi$$<... | KReiser | 21,412 | <p>In general, playing around with elements of a tensor product is something that should be avoided if you can help it (defining maps from a tensor product is also usually better done by defining them from the product and then checking that they are in fact $R$-balanced and the using the universal property). A proof wh... |
46,496 | <p>I'm tracing through solution for a question I was working on. I don't quite understand how they got to the line which I marked with an arrow (apologies for using an image, I don't have much time left and didn't wanna have to look up how to do sigma notation in LaTeX here)</p>
<p><img src="https://i.stack.imgur.com/... | Luffy | 12,342 | <p>You should use that $7=4+3$. If you look carefully, you can see that the line preceding your line with question marks should read something like $$7\sum_{k=3}^{n-1} \frac{1}{k} -3\sum_{k=3}^{n-1} \frac{1}{k} - 4 \sum_{k=3}^{n-1} \frac{1}{k} + 7(1+1/2) - 3(1/2+1/n) -4(1/n+1/(n+1)) = $$ $$ = 0 + 7(1+1/2) - 3(1/2+1... |
3,214,136 | <p>I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are <span class="math-container">$\sqrt{5}$</span>, <span class="math-container">$\sqrt{10}$</span>, <span class="math-container">$\sqrt{13}$</span>. After some algebra hell trying to work with Heron's formula, I plugged the qu... | Jerry Chang | 518,781 | <p><strong>Hint:</strong>
Observe that <span class="math-container">$5 = 1^2 + 2^2, 10 = 1^2+3^2$</span> and <span class="math-container">$13 = 2^2 + 3^2,$</span>
<a href="https://i.stack.imgur.com/n5uFI.png" rel="noreferrer"><img src="https://i.stack.imgur.com/n5uFI.png" alt="Square"></a></p>
|
127,779 | <p>I am doing an experiment to prove the associativity of the addition of points on an elliptic curve. So far, I have produced a code which allows me to move points on my curve. </p>
<p>To find their sum, I need to draw a line till it intersects a curve in the third point, then mirror this point around $x$ axis and pu... | J. M.'s persistent exhaustion | 50 | <p>Here's a starting point:</p>
<pre><code>ecp = ContourPlot[y^2 == x (x - 1) (x + 1), {x, -2, 2}, {y, -2, 2}];
ec = RegionNearest[ImplicitRegion[y^2 == x (x - 1) (x + 1), {{x, -2, 2}, {y, -2, 2}}]];
DynamicModule[{pts = {{-1, 0}, {1, 0}, {0, 0}}},
Panel[Row[{LocatorPane[Dynamic[pts, (pts =
... |
127,779 | <p>I am doing an experiment to prove the associativity of the addition of points on an elliptic curve. So far, I have produced a code which allows me to move points on my curve. </p>
<p>To find their sum, I need to draw a line till it intersects a curve in the third point, then mirror this point around $x$ axis and pu... | Jskud | 52,950 | <p>The fine solution offered by J. M. above works in Mma version 11, but not in 10.1 -- it uses the newer semantics for DistanceMatrix. To also work in an earlier version, you can use the following solution, using Complement[] instead of DistanceMatrix[]. (My newbie (low) reputation won't let me post comments, hence ... |
1,400,436 | <p>This is a question we asked on a second semester calculus test.</p>
<p>For what values of $p$ does this series converge?
$$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$</p>
<p>I believe that it actually can be shown that $p> 0$ is a valid answer. </p>
<p>However. I am interested in finding a proof that is simple ... | FundThmCalculus | 153,550 | <p>Comparison test:</p>
<ol>
<li>Note that $$-1 \leq \sin(x) \leq 1 ~~ \forall ~x$$</li>
<li>Therefore, $$\dfrac{\sin(x)}{f(n)} \leq \dfrac{1}{f(n)} ~~ \forall ~x$$</li>
</ol>
<p>So for your case: $f(n)=n^p$. This converges by $p$-series when $p>1$. Therefore, your original series converges for the same.</p>
<p>Y... |
4,180,865 | <p>A family of subsets <span class="math-container">$\mathcal{G}$</span> is called intersecting if <span class="math-container">$G_{1} \cap G_{2} \neq \emptyset$</span> for all <span class="math-container">$G_{1}, G_{2} \in \mathcal{G}$</span>. Let <span class="math-container">$\mathcal{F}_{1}, \mathcal{F}_{2}, \ldots,... | bpdolson | 958,587 | <p>Here is a way to see that <span class="math-container">$|\mathcal{F}| \le 2^{n-1}$</span> without invoking Erdős-Ko-Rado. If <span class="math-container">$\mathcal{F}$</span> contains more than half of all possible sets, it must contain some set and its complement, by the pigeonhole principle. But no set intersects ... |
1,591,706 | <p>Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$?</p>
<p>It looks like it might be possible to express this in terms of gamma functions and use Stirling's approximation.</p>
<p>It is clear that $\frac{an(an-1)...(an-n+1)}{n!}=\frac{\Gamma(an+1)}{\Gamma(a... | AAK | 517,170 | <p>I'll try to present the raw idea. The quantity changes sign frequently. Let us take it by modulus. For simplicity, let us assume that <span class="math-container">$a$</span> is an irrational number. Thus, we have
<span class="math-container">$$
S_n:=\left|\frac{an(an-1)...(an-n+1)}{n!}\right|=\frac{n^n}{n!}\prod_{0... |
269,474 | <p>If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that</p>
<p>$$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$</p>
<p>Also show when equality occurs.</p>
| Sugata Adhya | 36,242 | <p>Recall <a href="http://www.proofwiki.org/wiki/Linear_Transformations_Isomorphic_to_Matrix_Space" rel="nofollow">Linear Transformations Isomorphic to Matrix Space</a>.</p>
<p>Using <a href="http://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem" rel="nofollow">Rank–nullity theorem</a>, $\operatorname{rank}(A)+\op... |
2,942,761 | <p>Given these two examples from my math course:</p>
<p><strong>Example A</strong>:
<span class="math-container">$$\log(50) + \log(x/2) = 2 \implies \log 25x = 2 \implies 25x = 10^2 \implies x = 4.$$</span></p>
<p><strong>Example B</strong>:
<span class="math-container">$$\log(72) - \log(2x/3) = 0 \!\implies\! \log(... | chilean | 600,536 | <p>That function has not an elementary primitive. Although, you can integrate over <span class="math-container">$\mathbb R$</span>. Are you sure you didn't miss the interval of integration?</p>
|
2,942,761 | <p>Given these two examples from my math course:</p>
<p><strong>Example A</strong>:
<span class="math-container">$$\log(50) + \log(x/2) = 2 \implies \log 25x = 2 \implies 25x = 10^2 \implies x = 4.$$</span></p>
<p><strong>Example B</strong>:
<span class="math-container">$$\log(72) - \log(2x/3) = 0 \!\implies\! \log(... | Robert Israel | 8,508 | <p>Integration by parts works if you happen to know an antiderivative of <span class="math-container">$e^{x^2}$</span> (see the <a href="http://mathworld.wolfram.com/Erfi.html" rel="nofollow noreferrer">erfi function</a>).</p>
|
3,005,589 | <p>Let <span class="math-container">$R$</span> be a <em>commutative</em> ring with <span class="math-container">$1_R$</span>.</p>
<p>I found the following theorem as an equivalent definition of <span class="math-container">$R-$</span>modules.</p>
<blockquote>
<p><strong>Theorem.</strong> An abelian group <span clas... | Chris | 326,865 | <p><em>Let me give a try to answer and improve the theorem. Please feel free to edit my answer.</em></p>
<p><strong>Theorem.</strong> Let <span class="math-container">$R$</span> be a ring with <span class="math-container">$1_R$</span> and <span class="math-container">$(M,+)$</span> an abelian group (<span class="math-... |
250,397 | <p>If $(X,\mathcal{M})$ is a measurable space such that $\{x\}\in\mathcal{M}$ for all $x\in$$X$, a finite measure $\mu$ is called continuous if $\mu(\{x\})=0$ for all $x\in$$X$.</p>
<p>Now let $X=[0,\infty]$, $\mathcal{M}$ be the collection of the Lebesgue measurable subsets of $X$. Show that $\mu$ is continuous if an... | icurays1 | 49,070 | <p>It seems like the contrapositive is a good way to go. Suppose that $x\mapsto\mu([0,x])$ is not continuous, say at the point $x_0$. Then there exists an $\epsilon>0$ such that for all $\delta>0$ there is a $y$ such that $\vert x_0-y\vert<\delta$ but $\vert\mu([x_0,y])\vert\geq\epsilon$. Thus we can constr... |
2,214,287 | <p>My exam review states that I need to utilize the difference formula for sine to solve the equation on the interval $0 \leq \theta < 2\pi $</p>
<p>$$\sqrt3\sin \theta- \cos\theta = 1$$</p>
<p>I know that: $\sin \frac\pi3 = \frac{\sqrt3}{2}$
and $\cos\frac\pi3 = \frac12 $, so I divide each term by 2 and rewrite t... | Narasimham | 95,860 | <p>$$\sqrt3\sin \theta- \cos \theta = 1$$</p>
<p>$$2(\frac{\sqrt3}{2} \sin \theta- \frac{1}{2}\cos \theta)=1$$</p>
<p>$$(\frac{\sqrt3}{2} \sin \theta- \frac{1}{2}\cos \theta)=\frac12$$</p>
<p>$$ (\cos \dfrac{\pi}{6}\sin \theta -\sin \dfrac{\pi}{6} \cos \theta) = \sin \dfrac{\pi}{6} $$</p>
<p>$$\sin\left(\theta-\dfr... |
686,536 | <p>Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a proof. I need hints. Thank you.</p>
| Yiorgos S. Smyrlis | 57,021 | <p>In fact $a_nb_n\in L^1(\Omega)$ not $L^2(\Omega)$, and
$$
\int_\Omega a_nb_n\,dx \to \int_\Omega ab\,dx.
$$
Indeed,
$$
a_nb_n-ab=a_n(b_n-b)+(a_n-a)b
$$
Then
$$
\Big|\int_\Omega a_n(b_n-b)\,dx\,\Big|\le
\|a_n\|_{L^2}\|b_n-b\|_{L^2}\le M\|b_n-b\|_{L^2}\to 0,
$$
and
$$
\int_\Omega (a_n-a)b \to 0,
$$</p>
<h2>as $a_... |
2,301,450 | <p>Can anyone give a brief proof or a reference of a proof for the following property of <a href="https://en.wikipedia.org/wiki/Hilbert_space" rel="nofollow noreferrer">Hilbert spaces</a>?</p>
<blockquote>
<p>If <span class="math-container">$H$</span> is a Hilbert space and <span class="math-container">$M$</span> is... | ADA | 235,471 | <p>I will show you that if $X$ is Banach and Y is a closed subspace then $X/Y$ is Banach. From this hopefully you can extrapolate to your case. Now, you just need to define an inner product on your factor space.</p>
<p>To that end, let $X$ be Banach and $Y$ a closed subspace, and let $\{X_n\} \subset X/Y$ be an absolu... |
3,364,447 | <p>Let <span class="math-container">$\mathbb{F}_{p}$</span> be a finite field of order <span class="math-container">$p$</span> and <span class="math-container">$H_{n}(\mathbb{F}_{p})$</span> be the subgroup of
<span class="math-container">$GL_n(\mathbb{F}_{p})$</span> of upper triangular matrices with a diagonal
of one... | Nourddine Snanou | 550,778 | <p>The centre consists of upper-triangular matrices whose nonzero entries off the main diagonal are at the right upper corner. See Exercise 4. p 95 of (M. Suzuki, Group theory I, Springer Verlag, Berlin, 1982).</p>
|
370,570 | <p>In the wikipedia page (<a href="http://en.wikipedia.org/wiki/Birthday_problem" rel="nofollow">http://en.wikipedia.org/wiki/Birthday_problem</a>) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some pair of them will have the same birthday. ... | hejseb | 70,393 | <p>"Randomly chosen" concerns independence here. Later in the same article, </p>
<blockquote>
<p>"When events are independent of each other, the probability of all of the events occurring is equal to a product of the probabilities of each of the events occurring."</p>
</blockquote>
<p>That is the rationale for the ... |
1,469,695 | <p><a href="https://i.stack.imgur.com/fLWrM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLWrM.png" alt="enter image description here"></a></p>
<p>I have the following graph, and i'm trying to determine the maximum. When I use conventional methods, I end up with the ordered pair (2.35, 1.28) roug... | fleablood | 280,126 | <p>Do what they say. You have $k\mid r$ and $d\mid a$ so you have $kd\mid ar$. You have $k\mid s$ and $d\mid b$ so you have $kd\mid bs$ so you have $kd\mid ar + bs = d$. So you have $kd\mid d$.</p>
<p>Thus $k\leq 1$, which means $1$ is the largest number that divides both $r$ and $s$ so $\gcd(r,s) =1$.</p>
|
209,293 | <p>I'm trying to solve a set of 5 nonlinear equations using NSolve:</p>
<pre><code>exp1 := y*E^(x - z) == 18 a*x + b*y
exp2 := E^(x - z) == 8 a*y + b (x + z)
exp3 := -y*E^(x - z) == 72 a*z + b*y
exp4 := 9 x^2 + 4 y^2 + 36 z^2 == 36
exp5 := x*y + y*z == 1
NSolve[{exp1, exp2, exp3, exp4, exp5}, {x, y, z, a, b}, Reals]
<... | Bill | 18,890 | <p>Try</p>
<pre><code>exp1 = y*E^(x - z) == 18 a*x + b*y;
exp2 = 18 a*x + b*y == y*(8 a*y + b (x + z));
exp3 = 0 == 18 a*x + 72 a*z +2 b*y;
exp4 = 9 x^2 + 4 y^2 + 36 z^2 == 36;
exp5 = x*y + y*z == 1;
Reduce[{exp1, exp2, exp3, exp4, exp5}, {x, y, z, a, b}]
</code></pre>
<p>which adds some of your equations to others ... |
209,293 | <p>I'm trying to solve a set of 5 nonlinear equations using NSolve:</p>
<pre><code>exp1 := y*E^(x - z) == 18 a*x + b*y
exp2 := E^(x - z) == 8 a*y + b (x + z)
exp3 := -y*E^(x - z) == 72 a*z + b*y
exp4 := 9 x^2 + 4 y^2 + 36 z^2 == 36
exp5 := x*y + y*z == 1
NSolve[{exp1, exp2, exp3, exp4, exp5}, {x, y, z, a, b}, Reals]
<... | chyanog | 2,090 | <pre><code>AbsoluteTiming[
xyz = Reduce[{exp1, exp2, exp3, exp4, exp5, Element[{x, y, z}, Reals]}, {x, y, z}, {a, b}];
ans = NSolve[{exp1, exp2, exp3, xyz}, {x, y, z, a, b}] // Quiet;
]
ans
{exp1, exp2, exp3, exp4, exp5} /. ans
</code></pre>
<p>Output</p>
<blockquote>
<p>{0.466927, Null}<br>
{{x -> 0.222444,... |
1,439,337 | <blockquote>
<p>Let $(a_n)_n$ and $(\sigma_n^2)_n$ be sequences of real numbers with $\sigma_n^2>0$ for all $n\in \Bbb N$ and let $$\mathcal P = \{\Bbb P^{X_n} : X_n \sim \mathcal N (a_n,\sigma_n^2)\}.$$
Then $\mathcal P$ is tight if and only if there is a $K>0$ such that $|a_n|\leq K$ and $\sigma_n^2\leq K$ ... | Dominik | 259,493 | <p><strong>Hint:</strong> Use the fact that $X_n \stackrel{d}{=} \sigma_n X + a_n$, where $X$ is a standard Gaussian.</p>
|
2,038,245 | <p>So, an integral is notated like this:</p>
<p>$$\int_a^bf(x)dx$$</p>
<p>And from my understanding, it's an operator that is defined for three operands: $a$ and $b$, which can be anything, and an integrand of the form $f(x)dx$.</p>
<p>$dx$ is just an infinitesimal number, so $f(x)dx$ is simply $f(x)$ multiplied by ... | Mark Fischler | 150,362 | <p>Writing $$\sin z = \frac{e^{iz}-e^{-iz}}{2i} =
-\frac{i}{2}\left(e^{-y}(\cos x + i\sin x)-e^y (\cos x -i\sin x)\right)\\
=\sin x + \frac{i}{2} \cos x \left( e^y-e^{-y} \right)
$$
For the imaginary part of $\sin z$ to be zero, either:</p>
<ul>
<li><p>$y=0$ so that $e^{y}-e^{-y}=0$, in which case $z$ is real and $\si... |
932,907 | <p>So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p \rangle=e^{-ixp/\hbar}$. I can see that these are the eigenfunctions of position and momentum respectively, but I don't see how they'... | beep-boop | 127,192 | <p>No-one in their right mind would denote $\sin\left(x^2\right)$ as $\sin(x)^2$.</p>
<p>Why? Because the (round) brackets would become redundant. Brackets are used to remove ambiguity in algebraic operations. If you exclude the exponent $\quad ^2 \quad $ from the brackets, you're implicitly saying that the $\quad ^2... |
3,493,871 | <p>The problem is the following:</p>
<p>Let <span class="math-container">$\mathcal{F}$</span> be a family of distinct proper subsets of {1,2,...,n}. Suppose that for every <span class="math-container">$1\leq i\neq j\leq n$</span> there is a unique member of <span class="math-container">$\mathcal{F}$</span> that contai... | Misha Lavrov | 383,078 | <p>This result is known as the <a href="https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)" rel="nofollow noreferrer">de Bruijn–Erdős theorem</a>. It is usually stated in terms of incidence geometries. An incidence geometry has:</p>
<ul>
<li>A set of objects called <em>points</em>.... |
1,182,429 | <p>Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels are 1/3 and 1/4 respectively. If I did not know about the concept of LCDs, how would I figure out how much water the... | Bill Dubuque | 242 | <p>The fractions $\ x = \dfrac{a}b,\ y = \dfrac{c}d\ $ are, by definition, the (unique!) solutions of </p>
<p>the equations $\ bx = a,\ dy = c.\ $ To find their sum $\ x+ y \ $ it suffices to find </p>
<p>some equation $\ k (x + y) = j,\ $ giving the fraction $\, x+y = \dfrac{j}k\ $ for their sum. </p>
<p>By scal... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| Hari Rau-Murthy | 68,932 | <p><strong>General fact:</strong> Let <span class="math-container">$A_{-1} \subset \ldots A_n \subset \ldots$</span> be a filtration of cellular inclusions of <span class="math-container">$CW$</span> complexes. (More generally, let this be a filtration of cofibrations). Then <span class="math-container">$A_n$</span> ... |
675,991 | <p>So here's the question I'm trying to answer:</p>
<blockquote>
<p>Suppose $p_n(x) = \sum_{k=1}^N a_k^{(n)} x^k$ is a sequence of polynomials such that $p_n \to f$ uniformly over $[0,1]$ for some function $f:[0,1] \to \mathbb{R}$. Prove that $f$ must itself be an $N^\text{th}$ degree polynomial.</p>
</blockquote>
... | Joel | 85,072 | <p>You may start by sampling the points $0/n, 1/n, ..., (n-1)/n$, and make a vector $$\left( \begin{array}{c} p_m(0) - f(0)\\p_m\left(1/N\right) - f(1/N)\\ \vdots \\ p_m\left(N/N\right) - f( N/N) \end{array} \right) = \left(\begin{array}{cccc} 1& 0& \cdots& 0\\ 1& (1/N)& \cdots & (1/N)^N\\ \vdot... |
2,244,198 | <p>I am trying to prove that the </p>
<p>$A_n=n\int_{1}^{+\infty}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$ $\Rightarrow$
$\lim_{n\to\infty}A_n=0$ .</p>
<p>In fact, by defining $f_n(x)=\color{red}{n}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$, it is straightforward t... | Sungjin Kim | 67,070 | <p>Splitting the integral might be a good start. </p>
<p>Given $\epsilon>0$, write
$$
A_n=\int_1^{\infty} = \int_{1}^{x_n} + \int_{x_n}^{\infty}= C_n+D_n.$$
The sequence $x_n$ will be determined later so that $x_n\rightarrow\infty$ as $n\rightarrow\infty$. </p>
<p>We have by substitution $t=1-e^{-x}$, $dt=e^{-x}... |
1,066,921 | <p>Solve the system of equations $x^2=y^3, x^y=y^x$ in positive real numbers.</p>
<p>Taking $\ln$ of the second equation, we have $\ln x/x=\ln y/y$. This function is increasing in $(0,e)$ and decreasing in $(e,\infty)$. For any value of $x\neq e$, we can find a unique value of $y$ such that $x^y=y^x$. But how can we f... | Community | -1 | <p>Notice its better to start with the expressing y in terms of x because here you have $x^2 = y^3$ if you start expressing $x$ in terms of $y$ you would have to take the square root which would involve +- so you can start something like this
$x^2 = y^3 if y = x^{2/3}. $ now substitute that into the other formula and ... |
171,038 | <p>After some computations I end up with the following expression:</p>
<pre><code>Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]]
</code></pre>
<p>where $z$ is actually complex.</p>
<p>For reasons I don't understand, <em>Mathematica</em> won't simplify this to $1$.</p>
<p>According to the <a href="http://referen... | Michael Seifert | 27,813 | <p>For $z = 4 \sqrt{2} \pi i$, we have
$$
\sqrt{e^{-z/4 \sqrt{2}}} = \sqrt{e^{-i\pi}} = \sqrt{-1} = i,
$$ and
$$
\sqrt{e^{z/4 \sqrt{2}}} = \sqrt{e^{i\pi}} = \sqrt{-1} = i.
$$ So the product of the two square roots is $-1$ in this one case. Note that in this case,
$$
\arg e^{-z/4 \sqrt{2}} \neq -\frac{\Im(z)}{4 \sqr... |
2,463,756 | <p>I'm trying to prove that $f(x)=x^2-6x-40$ is injective for $f: [3, \infty) \rightarrow [-49, \infty)$. Note that I cannot use calculus.</p>
<p>I tried letting $f(a)=f(b)$ and I arrived at $a^2-6a=b^2-6b$. Then I tried to find a solution for $a$ in terms of $b$ and a solution for $b$ in terms of $a$ and I got $a=\fr... | velut luna | 139,981 | <p>You've got
$b=3 \pm \sqrt{9+a^2-6a}=3\pm\sqrt{(a-3)^2}$</p>
<p>Since $a \ge 3$,</p>
<p>$$b=3\pm(a-3)=a$$
or
$$b=6-a$$</p>
<p>Since $b \ge 3$, and $a \ge 3$, we conclude that it's only possible when
$$b=a$$</p>
|
3,209,773 | <p>Is the following result true? Or Is there any known result of fractions like this?</p>
<p>Let <span class="math-container">$n$</span> be fixed.</p>
<blockquote>
<p>There are infinitely many integer solutions for <span class="math-container">$$\sum_{i=1}^n \frac{1}{x_i} = 0,$$</span> where <span class="math-conta... | lulu | 252,071 | <p>To see that there are only finitely many solutions with <span class="math-container">$n,k$</span> fixed:</p>
<p>There are only finitely many multi-sets of size <span class="math-container">$≤n$</span> we can draw from <span class="math-container">$\{-1,\cdots, -k\}$</span>. Let <span class="math-container">$s$</s... |
2,698,775 | <p>Let E be the cylindrical frame field
<span class="math-container">$E_1 = \cos\theta U_1 + \sin\theta U_2, E_2 = − \sin\theta U_1 + \cos\theta U_2, E_3 = U_3$</span></p>
<p>(a) Starting from the basic cylindrical equations <span class="math-container">$x = r \cos\theta, y = r \sin\theta, z = z$</span>, show that the... | Narasimham | 95,860 | <p>Considering <em>directly</em> Cylindrical/polar coordinates for an arc <span class="math-container">$ds$</span> the three <em>differentials</em> are, :</p>
<p>along <span class="math-container">$z$</span> direction <span class="math-container">$=dz$</span>, </p>
<p>helical arc component along circumference direct... |
723,563 | <p>In the <strong>AM-GM Inequality</strong>, how do I conclude that $G(a)= a_1^\frac{1}{n},...,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=...=a_n = U(a)$ and ($U(a)= \frac{a_1+...a_n}{n}$ is the arithmetic mean)? Since if the $a_i$'s are not equal, then $G$ can be increased by the following procedure described ... | Cameron Williams | 22,551 | <p>It should be clear to figure out how many even numbers there are between $100$ and $400$. If you can't see it immediately, consider the amount of even numbers between $0$ and, say, $20$ and draw some conclusions from that.</p>
<p>Then note that $400-100=300$.. This should give a good indicator of the number of numb... |
835,536 | <p>Can you help me with this limit? What do I have to do? I'm lost.</p>
<p>$$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$</p>
<p>The solution given is $\dfrac{1}{2}$.</p>
| PenasRaul | 156,384 | <p>Compare with the integral of $\frac{n}{x^2}$ from $n$ to $2n$, and manage the error of the approximation.</p>
|
4,027,927 | <p>I'm trying to find derivative of <span class="math-container">$\frac{\cos t-\sin t}{\cos t+\sin t}$</span> in a different way: there is a trick to find derivative of the form <span class="math-container">$\frac{ax+b}{cx+d}$</span>:
<span class="math-container">$$\left(\frac{ax+b}{cx+d}\right)'=\frac{ad-bc}{(cx+d)^2... | J.G. | 56,861 | <p>Your function is<span class="math-container">$$\frac{1-u}{1+u}=\frac{2}{1+u}-1$$</span>with <span class="math-container">$u:=\tan t$</span>, so by the chain rule its derivative is <span class="math-container">$\frac{-2\sec^2t}{(1+\tan t)^2}=\frac{-2}{(\cos t+\sin t)^2}$</span>.</p>
|
3,113,186 | <p>Is this a power series? My book defines power series <span class="math-container">$\sum\limits_{n=1}^{+\infty} a_n (x-x_0)^n$</span></p>
| Mike | 544,150 | <p>If <span class="math-container">$|x| < 1$</span> then <span class="math-container">$|n^nx^{n!}| < e^{2n \ln n - n! \ln (\frac{1}{x})} < 2^{-n}$</span>.</p>
<p>So the radius of convergence is 1. (but the series does not converge for <span class="math-container">$x = \pm 1$</span>. </p>
|
3,113,186 | <p>Is this a power series? My book defines power series <span class="math-container">$\sum\limits_{n=1}^{+\infty} a_n (x-x_0)^n$</span></p>
| zhw. | 228,045 | <p>Clearly the series diverges if <span class="math-container">$|x|>1.$</span> For <span class="math-container">$|x|<1,$</span> apply the ratio test:</p>
<p><span class="math-container">$$\frac{(n+1)^{n+1}|x|^{(n+1)!}}{n^{n}|x|^{n!}} = (n+1)(1+1/n)^n|x|^{(n+1)!-n!}.$$</span></p>
<p>Verify that the expression on... |
259,156 | <p>I am using a piecewise function to define the height of columns. My goal is to make a nice picture that illustrates the different heights of the columns using color. Below in the picture I have described what I have generated using code on the left and on the right I have illustrated what I am trying to achieve:</p>... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
</code></pre>
<p>A slight variation on kglr's answer to track the original variable names.</p>
<pre><code>Manipulate[
Module[{
func = {Sin, Cos, Tan}[[n]],
var = {x, y, z}[[n]]},
Plot[func[a*t], {t, 0, n*Pi},
PlotRange -> If[n == 3, {-7, 7}, {-1, 1}],
AxesLabel -... |
2,691,818 | <p>I know from integration that the answer is -4. However, I am messing something up somewhere while working through the Riemann sums. Going cross-eyed trying to find my mistake. I included the pertinent steps and skipped the details. I do have those; just didn't type them in. Can anybody help me out? </p>
<p>$$\in... | Mohammad Riazi-Kermani | 514,496 | <p>You want to find the Riemann Sum for $$\int_{-1}^{1}\left(3x^{2}-3\right)\mathrm{d}x$$</p>
<p>Your limits are from $-1$ to $1$.</p>
<p>Thus your partition points are $$-1+2i/n , \text { for } 1\le i\le n$$</p>
<p>Thus you should have $$\lim_{n\to \infty}\sum_{i=1}^{n}\left(3\left(-1+\frac{2i}{n}\right)^{2}-3\ri... |
1,610,201 | <p>Will someone please help me with the following problem?</p>
<blockquote>
<p>Calculate the volume bounded between $z=x^2+y^2$ and $z=2x+3y+1$. </p>
</blockquote>
<p>As far as I understand, I need to switch to cylindrical coordinates:
$(h,\theta, r)$. </p>
<p>The problem is, that I can't understand how to find th... | dineshdileep | 41,541 | <p>Hint:</p>
<p>Since, you have reduced it to the diagonal case, we can consider $A$ and $B$ to be diagonal matrices without loss of generality. Let $x=[x_1,\dots,x_N]$ be a vector containing the diagonal elements of $A$ which are distinct. Consider the $N\times N$ vandermonde matrix
\begin{align}
V=\begin{bmatrix}
1 ... |
204,405 | <p>By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that commutes with $T$ and such that $\psi^2=T$.</p>
<p>I read somewhere that it does not have one, but could not find a ... | Douglas Lind | 8,112 | <p>A detailed study of the <em>group</em> of automorphisms of a shift of finite type (including the 2-shift mentioned here) is contained in <em>The Automorphism Group of a Shift of Finite Type</em>, Trans. Amer. Math. Soc. <strong>308</strong> (1988), 71-114 by me, Mike Boyle, and Dan Rudolph. I believe that the lack ... |
457,790 | <p>$\Big\{ \lim\limits_{x \to a} f(x-a) = L\Big\} \Longleftrightarrow \Big\{ \lim\limits_{h \to 0} f(h) = L\Big\}$</p>
<p>True or false? </p>
<hr>
<p><strong>Edit:</strong> Many thanks to the people who provided answers for this question.</p>
<p>I was confused about the domain of $f(x-a)$. If $D$ is the domain of $... | Stephen Herschkorn | 27,997 | <p>Hint: $\alpha + \beta = -2a, \ \alpha \beta = 3a$. These determine $\alpha$ as a function of $\beta$.</p>
|
457,790 | <p>$\Big\{ \lim\limits_{x \to a} f(x-a) = L\Big\} \Longleftrightarrow \Big\{ \lim\limits_{h \to 0} f(h) = L\Big\}$</p>
<p>True or false? </p>
<hr>
<p><strong>Edit:</strong> Many thanks to the people who provided answers for this question.</p>
<p>I was confused about the domain of $f(x-a)$. If $D$ is the domain of $... | Kunnysan | 84,764 | <p>$$|\alpha|,|\beta|\le 1\implies(1-\alpha^2)(1-\beta^2)\ge 0$$$$\implies1-(\alpha+\beta)^2+2\alpha\beta+(\alpha\beta)^2\ge0$$$$\implies1-4a^2+6a+9a^2\ge 0$$$$\implies(a+1)(5a+1)\ge0$$ Thus , $a\ge-1/5$ or $a\le-1$. But, $-2<-2a=\alpha+\beta< 2\implies -1< a< 1$ also, $a^2\ge 3a$ gives us, $a\le 0$ or $a\g... |
17,669 | <p>I just inherited two slide rules from my grandfather-in-law, one wood with smooth action despite nearly a century without
use.<sup>1</sup> (I used a K+E slide rule myself as an undergraduate in the 1970's.) It struck me that the conversion of multiplication/division to addition/subtraction via the
logarithmically ru... | Gerald Edgar | 127 | <blockquote>
<p>but I wonder if those who have, ever bring in to the classroom slide rules as "props"?</p>
</blockquote>
<p>In the Olden Days (before hand-held calculators) I remember that we had (at Ohio State) a big demonstration slide rule. It was maybe 6 feet long, on a stand with wheels. So the instructor wou... |
952,453 | <p>A fast food restaurant offers customer a choice of eight toppings that can be added to a hamburger. How many different hamburgers can be ordered?</p>
<p>Attempt: I don't know if this is correct 8!? I think there is no sufficient information. Can anyone please help me? Thank you.</p>
| mathlove | 78,967 | <p>If $Z\not =-1$, then$$\frac{X-Y}{Y}=Z\Rightarrow X-Y=YZ\Rightarrow X=Y(Z+1)\Rightarrow Y=\frac{X}{Z+1}.$$
Note that $X\not=0$ because $Y\not=0$.</p>
<p>If $Z=-1$, then
$$\frac{X-Y}{Y}=Z\iff X=Y(Z+1)=0\ \text{and}\ Y\not=0.$$
Hence, $Y$ can be any real number except $0$.</p>
<p>As a result, since $X=0\iff Z=-1$,
... |
2,968,241 | <blockquote>
<p>Let</p>
<p><span class="math-container">$$f(z)=\frac{e^{\sin{z}}-1}{z^3}.$$</span></p>
<p>a) Determine if <span class="math-container">$f$</span> has a pole at <span class="math-container">$0$</span> and determine its order.</p>
<p>b) If <span class="math-container">$f(z)=\sum_{n=-\inft... | Connor Harris | 102,456 | <p>(a) Near <span class="math-container">$z = 0$</span>, <span class="math-container">$\sin z \approx z$</span>, so <span class="math-container">$e^{\sin z} \approx 1 + z$</span>, so <span class="math-container">$\frac{e^{\sin z} - 1}{z^3} \approx \frac{1}{z^2}$</span>.</p>
<p>(b) The Taylor series of <span class="mat... |
340,417 | <p>How to calculate following integration?</p>
<p><span class="math-container">$$\int 5^{x+1}e^{2x-1}dx$$</span></p>
| Adi Dani | 12,848 | <p>$$\int 5^{x+1}e^{2x-1}dx=\int\frac{5}{e}(5e^2)^xdx$$</p>
<p>$$(5e^2)^x=u\Rightarrow(5e^2)^xdx=\frac{du}{\ln(5e^2)}$$</p>
<p>$$\int\frac{5}{e}(5e^2)^xdx=\frac{5}{e\ln(5e^2)}\int du=\frac{5}{e\ln(5e^2)}(u+C)$$</p>
|
340,417 | <p>How to calculate following integration?</p>
<p><span class="math-container">$$\int 5^{x+1}e^{2x-1}dx$$</span></p>
| MathGeek | 68,401 | <p><a href="http://www.wolframalpha.com/input/?i=%E2%88%AB5%5E%28x%2B1%29%2ae%5E%282x%E2%88%921%29dx" rel="nofollow">A computerised calculation of your problem</a></p>
|
2,617,576 | <p>I'm trying to check whether or not this set is linearly independent for all $n$, where $A$ is $n \times n$ and $A, A^2, \dots, A^{n^2}$ are distinct matrices and $I_n$ is the identity matrix.</p>
<p>Clearly, if we take $n = 2$, and $A = 3 I_n$ then the set $\{I, A, A^2, A^3, A^4 \}$ is not linearly independent. Is ... | Fred | 380,717 | <p>The vector space of $n \times n$ -matrices has dimension $n^2$, the set $\{I_n, A, A^2, \cdots, A^{n^2} \}$ has $n^2+1$ elements.</p>
<p>Conclusion ?</p>
|
113,362 | <blockquote>
<p>Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?</p>
</blockquote>
<p>Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later di... | Georges Elencwajg | 3,217 | <p>I'll just give one example showing that the distinction is indeed not absolute. </p>
<p>Traditionally in differential geometry you define tangent vectors to a manifold $V$ at a point $P$ first, for example as the vector space $T_P(V)$ of equivalence classes of differentiable curves through $P$ .<br>
And then you d... |
2,378,004 | <p>Let $n\ge 1$ be an integer and let $\vec{x} := \left( x_j \right)_{j=1}^n$ be normal variables with zero mean and with a correlation matrix ${\bf C}$. The question is to compute the following expectation value:
\begin{equation}
\mu_T(n):=E\left[ x_1 \cdot \prod\limits_{\xi=2}^n \theta_{T}(x_\xi) \right] = ?
\end{equ... | Przemo | 99,778 | <p>Here we provide an answer for $n=4$. We will be performing exactly the same steps as we did in the main body of the question. We have:
\begin{eqnarray}
&&\mu_T(4)=
\int\limits_{{\mathbb R}^4} x_1 \cdot \prod\limits_{\xi=2}^4 \theta_T(x_\xi) \cdot \frac{\exp\left[-\frac{1}{2} \vec{x}^T \cdot {\bf C}^{-1} \cd... |
92,105 | <p>Trying to solve</p>
<blockquote>
<p>$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.</p>
</blockquote>
<p>I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$</p>
<p>Would appreciate your help!</p>
| David Mitra | 18,986 | <p>Suppose $$\tag{1}\lim\limits_{x\rightarrow\infty}f(x)\ne 0.$$ </p>
<p>Then we may, and do, select an $\alpha>0$ and a sequence $\{x_n\}$ so that for any $n$, $$\tag{2}x_n\ge x_{n-1}+1$$
and
$$\tag{3}|f(x_n)|>\alpha.$$ </p>
<p>Now, since $f$ is uniformly continuous, there is a $1>\delta>0$ so that
$... |
65,355 | <p>In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a construction of a cluster category given a quiver with potential whose Jacobian algebra is finite dimensional (in parti... | Hugh Thomas | 468 | <p>Jan's answer includes many excellent references. I will try to give a few quick comments.</p>
<p>First of all, although the original Buan-Marsh-Reineke-Reiten-Todorov paper contained some results which were restricted to finite type cluster algebras, in subsequent work of them and others, notably Caldero-Keller, i... |
2,615,825 | <p>What I did was:
I tested for $\lim_\limits{n\to\infty}u_n$ by taking log</p>
<p>$$\lim_\limits{n\to\infty} \frac{\ln\ \left(4 - \frac{1}{n}\right)} {\frac{n}{(-1)^n}}$$</p>
<p>Applying L'hopital's rule,</p>
<p>$$\lim_\limits{n\to\infty} \frac{\left(\frac{1} {4-\frac{1}{n}}\right)\left(\frac{1}{n^2}\right)}{(-1)^n... | user8277998 | 516,330 | <p>The ellipse we have is in the form $S(x, y) = ax^2 + by^2 + 2hxy + c$. To remove the $xy$ term we rotate the ellipse by the angle $\tan (2\theta) = \dfrac{2h}{a - b}$. Which is same as rotating coordinate system by $-\theta$.</p>
<p>So if $(X', Y')$ is coordinates in new coordinate system then we can use</p>
<p>$$... |
2,812,118 | <p>Followup to the accepted answer of this question <a href="https://math.stackexchange.com/questions/631042/direct-proof-of-empty-set-being-subset-of-every-set">Direct proof of empty set being subset of every set</a></p>
<p>I understood the answer based on the nature of vacous truth, however what if we verify a state... | theyaoster | 356,200 | <p>If I've understood correctly, you are concerned about the statement $$\forall A, \forall x \in \varnothing : x \notin A.$$</p>
<p>Indeed, since $\forall x \in \varnothing$ is a statement about nothing, the statement is vacuously true. However, it does <em>not</em> mean that $\forall x : x \notin A$. In fact, the st... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | DonAntonio | 31,254 | <p>So basically you want to show that</p>
<p>$$\Bbb C[t]\ncong \Bbb C[x,y]/\langle x^2-y^3\rangle$$</p>
<p>I think your approach is good and you're pretty close: suppose there's an isomorphism $\,\phi\,$ , and let $\,p,q\in\Bbb C[t]\,$ be s.t. $\,\phi(p)=x+I\;\;,\;\phi(q)=y+I\,$ , with $\,I:=\langle x^2-y^3\rangle\,$... |
4,020,261 | <p>From my university notes:</p>
<blockquote>
<p><strong>Comment to slide 9</strong></p>
<p>By virtue of the result shown in this slide, we can talk about <em>the</em> least element of a set <span class="math-container">$D$</span>, if one exists, and we denote it with <span class="math-container">$\perp$</span>, pronou... | J.G. | 56,861 | <p><a href="https://detexify.kirelabs.org/classify.html" rel="nofollow noreferrer">Detexify</a> identifies it as <code>\sqsubseteq</code>. As discussed <a href="https://math.stackexchange.com/questions/1569400/does-sqsubset-have-any-special-meaning">here</a>, its meaning is highly context-dependent. But @MauroALLEGRANZ... |
1,607,395 | <blockquote>
<p>Consider quadratic equations <span class="math-container">$Ax^2 + Bx + C = 0$</span> in which <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, and <span class="math-container">$C$</span> are
independently distributed <span class="math-container">$\mathsf{Unif}(0,1)$<... | Jack D'Aurizio | 44,121 | <p>If $B$ is uniformly distributed over $[0,1]$ and $X=B^2$, the pdf of $X$ can be computed through:</p>
<p>$$\mathbb{P}[X\leq t] = \mathbb{P}[B\leq\sqrt{t}], $$
from which $f_X(x)=\frac{\mathbb{1}_{(0,1)}(x)}{2\sqrt{x}}$. In a similar way, if $A,C$ are uniformly distributed over $(0,1)$, independent, and $Y=AC$,
$$\m... |
1,997,050 | <p>Because strange conjectures have the tendency to enter my mind, I've become convinced that the only powers of $10$ that take the form $\frac{k(k+1)}{2}$ for $k\in \mathbb{Z}^{+}$ (i.e. is triangular) are $10^{0}$ and $10^{1}$. However, I'm having a difficult time making progress towards how to prove or disprove this... | Sarvesh Ravichandran Iyer | 316,409 | <p>Suppose that <span class="math-container">$k(k+1) = 2 \times 10^n$</span>. Then, we can solve for <span class="math-container">$k$</span> in terms of <span class="math-container">$n$</span> via the quadratic formula:
<span class="math-container">$$
k = \frac{1}{2} (\sqrt{8 \times 10^n + 1}-1)
$$</span></p>
<p>Thus,... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Community | -1 | <p>Here is a standard algebraic proof. It suffices to show that if $L/\mathbb{C}$ is a finite extension, then $L=\mathbb{C}$. By passing to a normal closure we assume that $L/\mathbb{R}$ is Galois with Galois group $G$. Let $H$ be the Sylow-2 subgroup of $G$ and $M=L^H$.</p>
<p>By the Fundamental Theorem of Galois The... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Andrea Ferretti | 828 | <p>I have collected 15 proofs with different approaches, including all the proofs suggested here so far. They are available at</p>
<p><a href="https://github.com/andreaferretti/math-notes/blob/master/TFA.pdf" rel="noreferrer">https://github.com/andreaferretti/math-notes/blob/master/TFA.pdf</a></p>
<p>Unfortunately th... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Ilya Nikokoshev | 65 | <p>Pukhlikov has a <a href="http://www.mathnet.ru/php/journal.phtml?wshow=paper&jrnid=mp&paperid=6&year=1997&volume=1&issue=&fpage=85&lpage=89&option_lang=eng">proof using only real numbers</a> (the page in English, full text in Russian) of the fact that indecomposable elements in $\mat... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Anonymous | 3,777 | <p>There's a linear algebra proof by Harm Derksen: <a href="https://www.jstor.org/stable/3647746" rel="nofollow noreferrer">https://www.jstor.org/stable/3647746</a>. You can also find the article posted here: <a href="https://math.berkeley.edu/%7Eribet/110/f03/derksen.pdf" rel="nofollow noreferrer">https://math.berkele... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Donu Arapura | 4,144 | <p>Here's an extract of my post to sci.math.research from 2001. The proof definitely
uses a "sledgehammer method", but perhaps it has some pedagogical value. I have no
doubt that other people may have come up with similar arguments.</p>
<hr>
<p>Sketch: It suffices to check that any complex monic polynomial has
a ro... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Jeff Strom | 3,634 | <p>Not quite an answer, but relevant:</p>
<p>Eilenberg and Niven proved that every "polynomial" in the quaternions has a root (provided
it has only one term of highest degree). The trick is familiar: they show that such a polynomial is homotopic to $q\mapsto q^n$, which induces a map of degree $n$ on the one-point c... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Moe Hirsch | 16,671 | <p>Two very short proofs, mostly topological, that a nonconstant polynomial map $f:{\bf C} \to \bf C$ is surjective (joint work with Robert Palais):</p>
<p><strong>(1)</strong> Complex analysis shows that $f$ is an open map (images of open sets are open). A standard estimate, $|f(z)|\to\infty$ as $|z|\to\infty$, im... |
133,406 | <p>Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$. </p>
<p>Is there a natural mixed Hodge structure on its equivariant cohomology?</p>
<p>Is it pure if $X$ is smooth projective? </p>
<p>What if we ask the analogous question for $l$-adic equivar... | Reladenine Vakalwe | 23,907 | <p>In the $\ell$-adic setting a reference would be Laszlo-Olsson's papers on six operations for Artin stacks:</p>
<p><a href="http://arxiv.org/abs/math/0512097" rel="nofollow">http://arxiv.org/abs/math/0512097</a></p>
<p><a href="http://arxiv.org/abs/math/0603680" rel="nofollow">http://arxiv.org/abs/math/0603680</a><... |
1,053,738 | <p>I've been asked to prove that:
$$
\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}
$$
When $0<x<2\pi$.</p>
<p>I know there are many similar posts on this site, but using $\cos(kx)$ instead, that's why I created this post, I can't get to the $1/2\cot$ in this one.
Thanks!</p... | Community | -1 | <p><strong>Hint:</strong> $\Im (z*i)=\Re(z)$, and $\frac {1}{i}=-i$.</p>
<p>Also, use $\cos$ of sum formula.</p>
<p>Use that on one of the formulas you already wrote.</p>
|
2,251,501 | <p>How can I calculate the number of perfect cubes among the first $4000$ positive integers?</p>
<p>Is there any trick to solving such questions?</p>
| Prune | 266,785 | <p>If you live in the world of base-2 geekdom, simply note that $2^{12}$ a.k.a. $16^3$ is 4K, or 4096. This is obviously too large. The barest mental math estimation will verify that $15^3 < 4000$. Done.</p>
|
956,490 | <p>A 5 card hand is dealt from a well-shuffled deck of 52 poker cards. If the first two cards are the 10 of diamonds and the 10 of hearts, what is the probability of having been dealt a full?</p>
<p>This is what I did: </p>
<p>Let $A$ be the event the first two cards are the 10 of diamonds and the 10 of hearts.
We w... | Graham Kemp | 135,106 | <p>By "full" I take it you mean a full hand?</p>
<p>You don't need Baye's rule for this.</p>
<p>The probability of being dealt a full hand when it is given that you have been dealt those two tens is the probability of also being dealt (among the 3 other cards drawn from the remaining 50) either: </p>
<ul>
<li>anothe... |
4,512,068 | <p>In an <a href="https://arxiv.org/abs/math/0506319" rel="nofollow noreferrer">article</a> by Guillera and Sondow, one of the <a href="https://mathworld.wolfram.com/UnitSquareIntegral.html" rel="nofollow noreferrer">unit square integral</a> identities that is proved (on p. 9) is: <span class="math-container">$$\int_{0... | Quanto | 686,284 | <p><span class="math-container">\begin{align}
&\int_{0}^{1} \int_{0}^{1} \left( - \frac{\ln(xy)}{1-xy} \right)^{m}\ \overset{xy=t}{dy} \ dx \\
=& \int_0^1 \int_0^x \left( - \frac{\ln t}{1-t} \right)^{m}\frac1x \ dt\ dx
= \int_0^1 \int_t^1 \left( - \frac{\ln t}{1-t} \right)^{m}\frac1x \ dx\ dt\\
= &\int_0^1... |
1,960,862 | <blockquote>
<p>If $a,b,c>0$, Then prove that $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\geq a+b+c$$</p>
</blockquote>
<p>$\bf{My\; Try::}$ Using Cauchy- Schwarz Inequality</p>
<p>$$\frac{a^4}{abc}+\frac{b^4}{abc}+\frac{c^4}{abc}\geq \frac{a^2+b^2+c^2}{3abc}$$</p>
<p>Now How can i solve after that , Help re... | Michael Hoppe | 93,935 | <p>By definition a curve $c$ is parameterized by arc length if the length of $c$ between $s$ and $t$ equals $t-s$:
$$\int_s^t\|\dot c(x)\|\,dx=t-s.$$
So by the fundamental theorem of calculus we have $\|\dot c(t)\|=1$, i.e., the tangent vector has length $1$.</p>
|
1,978,313 | <p>Confusion regarding solving inequality <span class="math-container">$\log_2(x-1) \geq 3$</span>. Now i got answer to be <span class="math-container">$[9,\infty]$</span>. But i don't see why we have to take intersection with domain at the end? </p>
<p>Thanks</p>
| Enrico M. | 266,764 | <p>First of all, take the necessary condition</p>
<p>$$x-1 > 0$$</p>
<p>hence</p>
<p>$$x > 1$$
otherwise you cannot proceed.</p>
<p>Now, exponentiate in base $2$:</p>
<p>$$2^{\log_2(x-1)} \geq 2^3$$</p>
<p>$$x-1 \geq 8$$</p>
<p>$$x \geq 9$$</p>
<p>Hence the domain is given by $[9, +\infty)$</p>
|
1,978,313 | <p>Confusion regarding solving inequality <span class="math-container">$\log_2(x-1) \geq 3$</span>. Now i got answer to be <span class="math-container">$[9,\infty]$</span>. But i don't see why we have to take intersection with domain at the end? </p>
<p>Thanks</p>
| Doug | 291,735 | <p>This is a question of notation really. This also seems like a homework problem because you said that you got the answer to be [9, $\infty$] yet you don't see why you chose to include ] at the end?</p>
<p>There is a question thread that covers this quite extensively.</p>
<p><a href="https://math.stackexchange.com/q... |
200,670 | <p>I use groupings as below: </p>
<pre><code>Join[
Groupings[IntegerPartitions[3], {A -> {2, Orderless}, B -> {2, Orderless}}],
{x}]
</code></pre>
<p>Which generates: </p>
<blockquote>
<pre><code>{A[2, 1], B[2, 1], A[A[1, 1], 1], A[B[1, 1], 1], B[A[1, 1], 1], B[B[1, 1], 1], 3}
</code></pre>
</blockquote>
... | Tim Laska | 61,809 | <p>Not an answer, but Tanh also approaches the Sign solution as one sharpens the transition so maybe Sign is behaving appropriately.</p>
<pre><code>X = ParametricNDSolveValue[{x''[t] ==
1 - x[t] - μ Tanh[ m x'[t]] Abs[1 + x[t]], x[0] == 0,
x'[0] == 0}, x, {t, 0, 50}, {μ, m}]
plts = Table[
Plot[Evaluate@Tab... |
1,196,633 | <p>I have the following recursive sequences:</p>
<p>$x_n = x_{n-1} + 2y_{n-1} , x_1 = 1$</p>
<p>$y_n = y_{n-1} - x_{n-1}, y_1 = -1$</p>
<p>where $ x_n,y_n \in \mathbb{Z}$ </p>
<p>I have to show that for any $n$ neither $x_n$ or $y_n$ are equal to 0.</p>
| Brian M. Scott | 12,042 | <p>HINT: Note that $x_1\bmod 3=1$ and $y_1\bmod 3=2$.</p>
<p>Suppose that $x_{n-1}\bmod 3=1$ and $y_{n-1}\bmod 3=2$; then</p>
<p>$$x_n\bmod 3=(x_{n-1}+2y_{n-1})\bmod 3=(1+4)\bmod 3=2\;,$$</p>
<p>and</p>
<p>$$y_n\bmod 3=(y_{n-1}-x_{n-1})\bmod 3=(2-1)\bmod 3=1\;.$$</p>
<p>What happens when $x_{n-1}\bmod 3=2$ and $y_... |
798,606 | <p>One can block-diagonally embed a copy $H$ of the unitary group $U(n-1)$ into $U(n)$ by
$$A \mapsto \begin{bmatrix}\det(A)^{-1}&0\\0& A\end{bmatrix}.$$</p>
<p>According to a remark in the example section of Paul Baum's dissertation, the coset space $$U(n)/H \approx S^1 \times \mathbb{C}\mathrm{P}^{n-1}.$$
... | Tyler Holden | 68,577 | <p>I think I can get a related result <em>locally</em> but I'm not sure about the general case:</p>
<p>The generalized Hopf fibration yields a principal bundle $S^1 \to S^{2n-1} \to \mathbb{CP}^{n-1}$ so that <em>locally</em> we have $S^{2n-1} \cong S^1 \times \mathbb{CP}^{n-1}$. We also have the principal bundle $U(n... |
3,274,496 | <p>Please correct my thinking, if anything not make sense to you.</p>
<p>A vector in <span class="math-container">$R^n$</span> is nothing but an assemblage of its co-ordinate w.r.t. some basis in the form of <span class="math-container">$n \times 1$</span> matrix.</p>
<p>Statement: If set of vectors(w.r.t. some basi... | egreg | 62,967 | <p>I believe you're looking from a wrong point of view.</p>
<p>A vector is <em>independent</em> from bases. It's an abstract object belonging to some vector space. As soon as you fix a basis, you can consider the <em>coordinates</em> of this vector with respect to the basis. If the vector space has dimension <span clas... |
932,596 | <p>Evaluation of $\displaystyle \int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx$</p>
<p>$\bf{My\; Solution::}$ Given $\displaystyle \int\frac{1}{\sin^2 x\cdot (5+4\cos x)}dx = \int \frac{1}{(1-\cos x)\cdot (1+\cos x)\cdot (5+4\cos x)}dx$</p>
<p>Now Using Partial fraction for $\displaystyle \frac{1}{(1-\cos x)\cd... | RE60K | 67,609 | <p>Yes there exists another one(OOps PF!):
$$\int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx
\stackrel{t=\tan x/2}=\int\frac{t^6+3 t^4+3 t^2+1}{4 t^4+36 t^2}dx\\
= \int(t^2/4+128/(9 (t^2+9))+1/(36 t^2)-3/2)dx=...$$
Similiar to your method. Your's is best, why are you looking for other methods?</p>
|
932,596 | <p>Evaluation of $\displaystyle \int\frac{1}{\sin^2 x\cdot \left(5+4\cos x\right)}dx$</p>
<p>$\bf{My\; Solution::}$ Given $\displaystyle \int\frac{1}{\sin^2 x\cdot (5+4\cos x)}dx = \int \frac{1}{(1-\cos x)\cdot (1+\cos x)\cdot (5+4\cos x)}dx$</p>
<p>Now Using Partial fraction for $\displaystyle \frac{1}{(1-\cos x)\cd... | Quanto | 686,284 | <p>Decompose the integrand as follows<span class="math-container">\begin{align}\int\frac{1}{\sin^2 x\left(5+4\cos x\right)}dx
=& \ \frac19 \int 5\csc^2x-\frac{4\cos x}{\sin^2 x}- \frac{16}{5+4\cos x}\ dx\\
= &\ \frac19\bigg(-5\cot x+\frac4{\sin x} -\frac{32}3\tan^{-1}\frac{\tan\frac x2}3 \bigg)
\end{align}</sp... |
2,980,446 | <p>I have this problem and I really don't know, how to edit it to get some solution.
<span class="math-container">$\lim_{n \to \infty} \sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span></p>
<p>So my question Is what to do with <span class="math-container">$\sqrt{n} * [\sqrt{n+1} - \sqrt{n}]$</span>? </p>
| Peter Szilas | 408,605 | <p>Multiplying by <span class="math-container">$ \dfrac {\sqrt {n+1} +√n}{\sqrt{n+1}+√n}=1 $</span> yields :</p>
<p><span class="math-container">$a_n:= \dfrac{√n}{\sqrt{n+1}+√n}.$</span></p>
<p><span class="math-container">$\dfrac{√n}{2\sqrt{n+1}} \lt a_n \lt \dfrac{√n}{2√n}=1/2$</span></p>
<p><span class="math-cont... |
244,429 | <p>If I have the following data (x value is composition and y value is temperature):</p>
<pre><code>data = {{0, 54.61`}, {100, 57.26243979492134`}, {80,53.839874154239816`}, {50, 54.09456572258326`}, {24, 56.15393883162748`}}
</code></pre>
<p>Which plotted like this gives:</p>
<pre><code>ListPlot[List /@ data, Frame -&... | Daniel Huber | 46,318 | <p>Consider your data:</p>
<pre><code>data = Sort@{{0, 54.61`}, {100, 57.26243979492134`}, {80,
53.839874154239816`}, {50, 54.09456572258326`}, {24,
56.15393883162748`}};
ListLinePlot[data]
</code></pre>
<p><a href="https://i.stack.imgur.com/Brb8w.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... |
234,137 | <p>I have a long list consists of float numbers with some of them duplicated as shown below.</p>
<p><a href="https://i.stack.imgur.com/Qwz8I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qwz8I.png" alt="list output" /></a></p>
<p>I want to divide this list into bins;</p>
<ol>
<li>with equal bin wid... | azerbajdzan | 53,172 | <p><code>binamount</code> is equal:</p>
<pre><code>binamount == Floor[(max - min)/binwidth]
</code></pre>
<p>Therefore binwidth lies in this interval:</p>
<pre><code>(max - min)/(binamount + 1) < binwidth <= (max - min)/binamount
</code></pre>
<p>So for your original list:</p>
<pre><code>list = {1, 2, 2, 3, 4, 5,... |
1,334,557 | <p>Suppose that an entire function $f$ has uncountably many zeros. Is it true that $f=0$?</p>
<p>I have no idea how to proceed with this. Perhaps some theorem that I am not aware of. I have done an undergraduate course in complex analysis.</p>
| partha | 107,781 | <p>Lets consider open ball $B_n$ at center zero and radius $n$, a natural number, then $\mathbb{C}=\cup_{n\in N} B_n$. Now if none of $B_n$ contains uncountably many complex zeros of $f$, zeros of $f$ will become countable, a contradiction. So suppose $B_k$ for some natural number $k$, contains uncountably many comple... |
465,555 | <p>Let $f , g : X \rightarrow Y$ be continuous where $Y$ is Hausdorff. Prove that $A = \{x : f(x) = g(x)\}$ is closed in $X$.
I have done the followings.</p>
<p>$f(X)$ and $g(X)$ are two subspaces of $Y$.</p>
<p>As Y is Hausdorff, $f(X), g(X)$ and $f(X) \times g(X)$ are also.</p>
<p>$L = \{(f(X),g(X)) : f(X) = g(X)\... | Hagen von Eitzen | 39,174 | <p>$Y$ is Hausdorff iff the diagonal $\Delta\subseteq Y\times Y$ is closed. And $A=(f,g)^{-1}(\Delta)$.</p>
|
1,482,253 | <p>Prove the following. What would be the summation formula be for the first part?</p>
<p><a href="https://i.stack.imgur.com/nkO0A.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nkO0A.jpg" alt="enter image description here"></a></p>
| glebovg | 36,367 | <p>To prove (a) and (b) use the binomial theorem. To prove (c) use the hint, that is, bound $e_n$ above by a geometric progression with ratio $1/2$.</p>
|
3,988,517 | <p>Consider
<span class="math-container">$$A=a^2+2ab^2+b^4-4bc-4b^3,$$</span>
where <span class="math-container">$a,b,c\in\mathbb{Z}$</span> and <span class="math-container">$b\neq0$</span> such that <span class="math-container">$b|a$</span> and <span class="math-container">$b|c$</span>, so <span class="math-container"... | AugSB | 188,245 | <p>No, <span class="math-container">$A$</span> is not a perfect square for such <span class="math-container">$a,b,c$</span> values. As a counterexample, take <span class="math-container">$a=b=c=1$</span>, for which <span class="math-container">$A=-4$</span>.</p>
<hr />
<p><strong>EDIT:</strong> The answer below corresp... |
105,378 | <p>I want to ask you if can it be so simple to prove that $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ by divide the numerator and denominator with $x^2$ and that's it? </p>
<p>If it this simple indeed you can write a comment and I'll delete the question after I'll read it... | Robert Israel | 8,508 | <p>In general, interchanging limits and sums or integrals can be tricky. It's not always true that<br>
$\lim_{x \to \infty} \sum_{n=1}^\infty f_n(x) = \sum_{n=1}^\infty \lim_{x \to \infty} f_n(x)$, even when both sides converge. It is true for dominated
convergence (if there is some convergent series $\sum_{n=1}^\inf... |
3,446,693 | <p>A highway patrol plane is flying <span class="math-container">$1$</span> mile above a long, straight road, with constant ground speed of <span class="math-container">$120$</span> m.p.h. Using radar, the pilot detects a car whose distance from the plane is <span class="math-container">$1.5$</span> miles and decreasin... | N. F. Taussig | 173,070 | <p>You defined <span class="math-container">$x$</span> as the horizontal distance between the airplane and car. As the airplane approaches the car, the distance shrinks, which makes
<span class="math-container">$$\frac{dx}{dt}$$</span>
negative. </p>
<p>The rate at which the horizontal distance shrinks depends on wh... |
3,294,237 | <blockquote>
<p><span class="math-container">$\begin{array}{l}{\text { if } \operatorname{gcd}(a, n)=1 \quad \& \operatorname{gcd}(b, m)=1} \\ {a^{x} \equiv s \bmod (n)} \\ {b^{x} \equiv s \bmod (m)} \\ {\text { is there result relating }(a b)^{x} \text { with } \bmod (m n) ? ?}\end{array}$</span></p>
</blockquot... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> Specialize the following mixed modulus <span class="math-container">$ $</span> <em>Product Rule</em></p>
<p><span class="math-container">$\qquad \begin{align} x&\equiv s\!\!\pmod{\!n}\\ y&\equiv t\!\!\pmod{\!m}\end{align}\ $</span> <span class="... |
3,856,233 | <p>Let us start from probability space <span class="math-container">$(\Omega,\mathcal{C},\mathbb{P})$</span> and a sequence of events <span class="math-container">$\{C_n\}$</span>. I know that:
<span class="math-container">$$\mathbb{P}\left(\bigcup_{m\ge n}C_m\right)\ge\mathbb{P}\left(C_m\right),\text{ for each $m\ge n... | Community | -1 | <p>For each <span class="math-container">$x\in X$</span>, we have a sequence of zeros and ones. Meanwhile the Cantor set is the set of all real numbers in the unit interval whose ternary expansion contains no <span class="math-container">$1$</span>'s. So the natural map would be to send a given sequence <span class="... |
762,762 | <p>Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$</p>
<p>What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$?</p>
<p>The answer is $a_n$, but I can't solve it.</p>
<p>See, what I've done is substitute $x$ as $-\frac{1}{x}$ and I've got:</p>
<p>${\frac{(x^2-x+1)}{... | Mario Carneiro | 50,776 | <p>Let $(1+x+x^2)^n=\sum_ka_kx^k$. Then:</p>
<p>\begin{align}
(1+x^2+x^4)^n&=(1-x^{-1}+x^{-2})^n(1+x+x^2)^nx^{2n}\\
\sum_ja_jx^{2j}&=\sum_k(-1)^ka_kx^{-k}\sum_ja_jx^jx^{2n}\\
&=\sum_j\sum_k(-1)^ka_ka_jx^{2n+j-k}\\
&=\sum_j\sum_k(-1)^ka_ka_{k+j-2n}x^j\\
\end{align}</p>
<p>The $x^{2n}$ coefficient on th... |
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