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Top Questions
9
votes
A "binomial" generalization of harmonic numbers
binomial-coefficients
integer-partitions
harmonic-numbers
asked May 19, 2019 at 18:02
math.stackexchange.com
9
votes
Integrate $\int_a^\infty\frac{\sqrt{x^2-a^2}}{\sinh x}\,dx$
definite-integrals
special-functions
closed-form
asked Jul 8, 2019 at 18:53
math.stackexchange.com
6
votes
Number of independent values among $\{\Gamma(k/n):0<k<n/2\}$
elementary-number-theory
gamma-function
rational-numbers
asked Feb 12, 2021 at 19:42
math.stackexchange.com
5
votes
Behavior of $n\sqrt{3}-a_n\sqrt{n}$ as $n\to\infty$, where $a_{n+1}=a_n+n/(a_1+\dots+a_n)$
sequences-and-series
asymptotics
recurrences
asked Sep 2, 2022 at 8:15
mathoverflow.net
Top Answers
29
Prove that the quantity is an integer
math.stackexchange.com
26
Euler-Mascheroni constant in Bessel function integral
math.stackexchange.com
25
Integrate $I(n)=\int_{-1}^\infty \frac{\ln^n(x+1)}{x^n}\mathrm dx$
math.stackexchange.com
19
Prove $\sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right) \arctan\left(\frac{1}{F_{n+1}}\right)=\frac{\pi^2}{8}$
math.stackexchange.com
18
Integral of Binomial Coefficient
math.stackexchange.com
17
Evaluate $\lim\limits_{n \to \infty}\frac{(-1)^n}{n!}\int_0^n (x-1)(x-2)\cdots(x-n){\rm d}x$
math.stackexchange.com
17
How to evaluate $\int_{-\infty}^{\infty}e^{x-n\sinh^2 x}\ dx$
math.stackexchange.com
16
Shannon entropy of a fair dice
math.stackexchange.com
16
Analytic continuation of $\Phi(s)=\sum_{n \ge 1} e^{-n^s}$
math.stackexchange.com
15
Computing $\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$
math.stackexchange.com
15
Proving $\prod_{n=1}^{\infty}\left(1+\frac{1}{n^{3}}\right)=\frac{1}{\pi}\cosh\frac{\pi\sqrt{3}}{2}$
math.stackexchange.com
15
How to minimise the cost of guessing a number in a high/low guess game?
math.stackexchange.com
15
Maclaurin series of $1- \cos^{2/3} x$ has all coefficients positive
math.stackexchange.com
13
Is the following set dense in $L^2$?
math.stackexchange.com
13
Asymptotic estimation problem about $\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\frac{{i + j}}{{{i^2} + {j^2}}}} } $
math.stackexchange.com
12
Closed form or upper bound for $\lim_{n\to\infty}\sum_{k=0}^n\frac{e^{-k^2x}-e^{-(k+1)^2x}}{2k+1} $?
math.stackexchange.com
12
Limit of $\int_0^1\left(\frac {2}{\sqrt {(1-t^2)(1-xt^2)}}-\frac{x}{1-xt}\right)\,dt$ as $x\to 1^{-}$
math.stackexchange.com
12
Evaluating $\lim _{n \rightarrow \infty} \sum_{k=n+1}^{2 n} \sin \frac{n}{n^{2}+k^{2}}$
math.stackexchange.com
12
A prime numbers property: $4/\pi=e^{1/2}\Big(1+e^{1/3}\big(1+e^{1/5}(\dots)\big)\Big)/p_n, \ \ n\to\infty$
math.stackexchange.com
11
Is there a $f: \mathbb{N} \to \mathbb{N}$ such that $\sum_{n=1}^{\infty} \frac{1}{n^2f(n)} \in \mathbb{Q}$?
math.stackexchange.com
11
How to prove $\int_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}-\frac{\zeta'(2)}{\zeta(2)}+\gamma-\frac73\log(2)$?
math.stackexchange.com
10
Evaluate the summation $\sum_{n=1}^{10} n \left( \frac{1^2}{1 + n} + \frac{2^2}{2 + n} + ....+\frac{10^2}{10 + n}\right)$
math.stackexchange.com
10
A Wallis-like formula for $\pi$: $(\frac21)^2(\frac23)^2(\frac43)^2(\frac45)(\frac65)^2(\frac67)^2(\frac87)(\frac89)^2\cdots$
math.stackexchange.com
10
Evaluate limit of $a_n=\frac{n+1}{2^{n+1}}\sum\limits_{i=1}^{n}\frac{2^i}{i}$
math.stackexchange.com
10
On $\mathrm{\sum\limits_{x=1}^\infty Ci(x)}$.
math.stackexchange.com
9
Evaluate $\int_{(-\infty,\infty)^n}\frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k}\frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k}$
math.stackexchange.com
9
Proving $\int_0^\infty\left(\frac{x^xe^{-x}}{\Gamma(x+1)}-\frac1{\sqrt{2\pi x}}\right)dx=-\frac13$
math.stackexchange.com
9
Approximate result for $\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)$?
math.stackexchange.com
9
Evaluating $\int_{0}^{\frac{\pi}{2}} x^n \csc(x) dx$
math.stackexchange.com
9
Interesting patterns in $f(k,n)=k\pi-\sum\limits_{x=1}^n\tan^{-1}\left(\frac1{\sqrt[k]x}\right)$
math.stackexchange.com
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