Winther
  • Member for 7 years, 8 months
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Proof (claimed) for Riemann hypothesis on ArXiv
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51 votes

I had a go reading through the paper and I think I found the error. The main argument in the paper can be summarized as follows: The Riemann $\Xi$-function $\Xi(t) = \xi\left(\frac{1}{2} + it\right)...

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The expected outcome of a random game of chess?
31 votes

I found a bug in the code given in Hooked's answer (which means that my original reanalysis was also flawed): one also have to check for insufficient material when assessing a draw, i.e. int(board....

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Integration and differentiation of Fourier series
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30 votes

Fourier series (as with infinite series in general) cannot always be term-by-term differentiated. For general series we have the following theorem Theorem: Term-by-term differentiation: If a series ...

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An integral identity from Ramanujan's notebooks
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25 votes

It's a nice identity, but this one is fairly simple compared to some of the other magical results in the notebook. The function $f(a) = \int_0^\infty \frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}} \frac{...

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$f(f(f(x))) = x$. Prove or disprove that f is the identity function
17 votes

You can prove that $f(x) \equiv x$ in four fairly simple steps: Show that $f$ is one-to-one. Assume $f(x) = f(y)$ and show that this implies $x=y$ by applying $f$ two times to each side of the ...

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Intuition behind the paradox of instantaneous heat propagation
14 votes

First the wave-equation: let's start with a small narrow pulse located close to $x=0$ ($u$ nonzero close to $x=0$ and zero elsewhere) and then let it go at $t=0$. Now let's see how long time does it ...

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A Neat Identity Involving Zeta Zeroes
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14 votes

The result found by OP turns out to be quite generic; it holds for a wide range of sequences $\rho_n$ and not just the zeros of the $\zeta$-function. A precise formulation of what is observed is the ...

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An integral identity from Ramanujan's notebooks
14 votes

Here is another solution using the residue theorem. If we try to integrate the function $\frac{\sinh(az)}{\sinh(\pi z)}\frac{1}{1+n^2z^2}$ directly then we obtain the result $$\int_0^\infty \frac{\...

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Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
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14 votes

Make a change of variables $u = e^x$ $$\int_0^1 f(x) e^{nx}dx = \int_{1}^e g(u) u^{n}du = 0$$ where $g(u) = \frac{f(\log(u))}{u}$ is a continuous function. You can now apply Weierstrass ...

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How can I prove that $\int_{0}^{\infty }\frac{\log(1+x)}{x(1+x)}dx=\sum_{n=1}^{\infty }\frac{1}{n^2}$
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14 votes

The substitution $u = \log(1+x)$ gives $$I = \int_0^\infty \frac{\log(1+x)}{x(x+1)} dx = \int_0^\infty \frac{udu}{e^u-1}$$ Now by using $\frac{1}{1-x} = 1+x+x^2+\ldots$ with $x=e^{-u}$ we get $$I = ...

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The limit of $(n!)^{1/n}/n$ as $n\to\infty$
14 votes

Put $$a_n = \frac{n!^{1/n}}{n}$$ then $$\log a_n = \frac{1}{n}\left(\log n! - n\log n\right)= \frac{1}{n}\sum_{k=1}^n\log\left(\frac{k}{n}\right)$$ where we have used $\log n! = \log 1 + \log 2 + \...

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Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$
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14 votes

Just in case it is not assumed that $\alpha\in\mathbb{R}$, let $\alpha=a+bi$ with $a,b\in\mathbb{R}$. Then $$\begin{align} \cos(a\pi+b\pi i)&=\cos(a\pi)\sin(b\pi i)+\sin(a\pi)\cos(b\pi i)\\ \...

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How to prove this inequality $x^2_{n}\le\frac{8}{3}$
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12 votes

Just so this question is fully answered (i.e. getting the bound $8/3$) here I add a condensed proof of $x_n^2 \leq \frac{8}{3}$ from the answer given here as Community Wiki. Define $$E_n = (x_n - x_{...

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Could a square be a perfect number?
11 votes

A perfect square can be written on the form $n=(p_1^{\alpha_1}\cdots p_k^{\alpha_k})^2$ where $p_i$ are some prime numbers and $\alpha_i> 0$ are integers. The sum of the divisors of $n$ is given ...

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Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$
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11 votes

I will here derive an expression for the more general problem of evaluating $\sum_{n=1}^\infty \frac{H_n z^{2n+1}}{(2n+1)^2}$ for $|z| < 1$ in terms of polylogarithms. This derivation uses the same ...

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Let $f$ be a function such that $f'(x)=\frac{1}{x}$ and $f(1) = 0$ , show that $f(xy) = f(x) + f(y)$
11 votes

From the definition $f'(x) = \frac{1}{x}$ with $f(0)=1$ we have $$f(x) \equiv \int_1^x \frac{dt}{t}$$ By a simple change of variables $z = ty$ to the integral above we get the result $$f(x) = \int_y^...

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Equality of laplace transform
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11 votes

This is a theorem knows as Lerchs theorem. I will give a standard proof of this below. I will restrict myself to continuous functions, but the generalization to integrable functions (continuous almost ...

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Prove that $f(x)=\sum_\limits{n=1}^{\infty} \frac{1}{x^2+n^2}$ is differentiable on $\mathbb{R}$
Accepted answer
10 votes

Your approach looks fine. The theorem you need to justify your claims is the following (see for example Theorem 2 in this note): Term by term differentiation theorem: If $\sum f_k(x_0)$ converges at ...

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Bound on matrix product $\begin{bmatrix} 1+\frac{1}{n} & -1 \\ 1 & 0 \end{bmatrix}\cdots\begin{bmatrix} 1+\frac{1}{2} & -1 \\ 1 & 0 \end{bmatrix}$
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10 votes

TLDR; We can get $$x_n^2 \leq \frac{8}{3}$$ directly and with some computation we can establish the best bound $x_n^2 \leq \frac{9}{4}$ for all $n$. The argument below can be made much shorter, I just ...

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Prove that: $\zeta(3)=\lim_{N\to \infty}{1\over N}\sum_{k=1}^{N}{1\over k^{\phi^2}\ln{\left(1+{k^{\phi^{-2}}\over N}\right)}}$
Accepted answer
10 votes

Rough estimate: We have $\log(1+x) \approx x$ as $|x| \ll 1$ so for large $N$ the sum is approximately $$ \frac{1}{N}\sum_{k=1}^N \frac{1}{k^{\phi^2}\log\left(1 + \frac{k^{\phi^{-2}}}{N}\right)} \...

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Prove that $\exp(A+B)=\exp(A)\exp(B)$ iff $[A,B] = 0$
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10 votes

To show $e^{(A+B)t} = e^{At}e^{Bt}$ for all $t$ $\implies [A,B] = 0$ compare the coefficient of $t^2$ in the power-series expansion of both sides of the equation. We have $$e^{(A+B)t} = 1+ (A+B)t + \...

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Find $\lfloor 1000S \rfloor$ for $S =\sum_{n=1}^{\infty} \frac{1}{2^{n^2}} = \frac{1}{2^1}+\frac{1}{2^4}+\frac{1}{2^9}+\cdots.$
Accepted answer
10 votes

The sum of the first three terms is $$1000\left[\frac{1}{2^{1}}+\frac{1}{2^{4}}+\frac{1}{2^{9}}\right] \simeq 564.453$$ The sum of the rest of the terms is bounded by (using a geometrical series) $$...

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Proving that $\sum_{n=0}^{\infty }\frac{3(n!)^2}{(2n+2)!}=\sum_{n=1}^{\infty }\frac{1}{n^2}=\frac{\pi ^2}{6}$
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10 votes

Using the series expansion of $\arcsin^2(x)$ (see this question) given by $$\arcsin^2(x) = \frac{1}{2}\sum_{n=1}^\infty\frac{(2x)^{2n}}{n^2{2n\choose n}}$$ we see that your sum is $$\sum_{n=0}^\...

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Find $a_{n,i,j}$ in the expansion $(x + D)^n = \sum\limits_{i,j} a_{n,i,j} x^i D^j.$
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10 votes

The operator $e^{t(x+D)}$ is an exponential generating function for the sequence $(x+D)^n$. The commutator of the two operators $x$ and $D$ is given by $[x,D] = xD - Dx = -1$ and $[[x,D],x] = [[x,D],D]...

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Evaluating $\int_{-\infty}^\infty\frac1{1+x^2+x^4+\cdots}\ \text{dx}$
9 votes

This is a bit overkill for solving the question at hand, but for completness I'll add an answer for how one can evaluate the integral for a general value of $n$ using residue calculus giving us the ...

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Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?
Accepted answer
9 votes

Let's first start with defining what we mean by the expression $a = x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$: the infinite power tower is the limit of the recursion $$a_{n+1} = x^{a_n},~~~~~~~ a_0 = x$$ ...

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If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $
9 votes

This problem has an interesting history from the early days of real analysis attached to it. I really like this story and the proof it contains of the problem in this question is also different from ...

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Principal branch of logarithm
Accepted answer
9 votes

The complex logarithm $\log(z)$ is defined as the inverse function to the exponential function, i.e. it satisfies $e^{\log(z)} \equiv z$. Since $re^{i\theta} = re^{i\theta + 2\pi k i}$ for all $k\in\...

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Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
9 votes

Hint: A convex function always satisfy $$f(tx+(1-t)y)\leq tf(x) + (1-t)f(y),~~~~~~~t\in[0,1]$$ Take $f(x) = |x|^p$, show that this is convex for $p\geq 1$ (for example by the second derivative test)...

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How to find continued fraction of pi
8 votes

A neat method to construct a continued fraction for $\pi$ is to use the addition formula for $\arctan$: $$\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$ which can also be written $$...

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