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Ѕᴀᴀᴅ
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Proving $(λ^d + (1-λ^d)e^{(d-1)s})^{\frac{1}{1-d}}\leq\sum\limits_{n=0}^\infty\frac1{n!}λ^{\frac{(d^n-1)d}{d-1}+n}s^ne^{-λs}$

Prove $\lim\limits_{n\to∞}{\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-(x+1)})^{n+1}\over\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-x})^{n + 1}}=\frac 13$

Does $\sum\limits_{k=1}^n\frac{a_i-a_k}{a_i+a_k}\cdot\frac{a_j-a_k}{a_j+a_k}=0$ for all $i\neq j$ imply $a_1=a_2=\cdots=a_n$?

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30
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
Apr 16, 2018
20
Calculate a quartic monic polynomial given that $P(1)=10,P(2)=20$ and $P(3)=30$.
Apr 2, 2018
20
Proving $(λ^d + (1-λ^d)e^{(d-1)s})^{\frac{1}{1-d}}\leq\sum\limits_{n=0}^\infty\frac1{n!}λ^{\frac{(d^n-1)d}{d-1}+n}s^ne^{-λs}$
Jan 27, 2018
16
Maximum of $\int_a^b \frac{f(x)}{x}\,\mathrm dx$
Apr 24, 2019
14
Prove $\lim\limits_{n\to∞}{\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-(x+1)})^{n+1}\over\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-x})^{n + 1}}=\frac 13$
Mar 9, 2018
13
Prove $\sum\limits_{n=0}^\infty\frac{5^n(3^{5^{n+1}}-5\cdot3^{5^n}+4)}{(729)^{5^n}-(243)^{5^n}-5\cdot3^{5^n}+1}=\frac12$
Mar 3, 2018
12
How to compute the determinant of this Toeplitz matrix?
Dec 24, 2017
11
If $a+b+c<0$ and $ax^2+bx+c=0$ has no real roots, is it true that $c$ must be less than $0$?
Mar 10, 2018
11
Evaluating $\lim_{n \to \infty}\left(\sqrt{n^2 - n+1}-\left\lfloor\sqrt {n^2 - n+1}\right\rfloor\right)$
Oct 2, 2018
10
$x$-coordinate distribution on the $n$-sphere
Nov 24, 2018
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