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Ѕᴀᴀᴅ
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Series expansion of a function defined by a definite integral

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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View all questions and answers
All Questions Answers
Score Newest
answer
30
Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?
Apr 16, 2018
answer
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
answer
20
Calculate a quartic monic polynomial given that $P(1)=10,P(2)=20$ and $P(3)=30$.
Apr 2, 2018
answer
16
Maximum of $\int_a^b \frac{f(x)}{x}\,\mathrm dx$
Apr 24, 2019
answer
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
question
12
How to compute the determinant of this Toeplitz matrix?
Dec 24, 2017
answer
12
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
answer
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
answer
10
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
answer
10
$x$-coordinate distribution on the $n$-sphere
Nov 24, 2018
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