Question

An Euler type sum: $\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n\cdot 4^n}{2n \choose n}$, where $H_n^{(2)}=\sum\limits_{k=1}^{n}\frac{1}{k^2}$

score 438 · Answer 1 · 2026-03-25 03:03:18

438

Views

An Euler type sum: $\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n\cdot 4^n}{2n \choose n}$, where $H_n^{(2)}=\sum\limits_{k=1}^{n}\frac{1}{k^2}$

Published on 25 Mar 2026 - 3:03

score 993 · Answer 2 · 2018-08-27 00:39:37

993

Views

How do I convert a sum (from $1$ to $n$) of powers into a function of $n$?

Published on 27 Aug 2018 - 0:39

score 882 · Answer 3 · 2026-03-19 21:53:32

882

Views

Sum to n terms the series $\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$

Published on 19 Mar 2026 - 21:53

score 2.6k · Answer 4 · 2026-03-20 10:21:38

2.6k

Views

Sum to n terms of the following series: $2 \cdot 2 + 6 \cdot 4 + 12 \cdot 8 + 20 \cdot 16 + \cdots $

Published on 20 Mar 2026 - 10:21

score 76 · Answer 5 · 2018-08-27 22:14:16

76

Views

Show that $\ln\left(1+3x + 2x^2\right) = 3x - \frac{5x^2}{2} + \frac{9x^3}{3} -\cdots + \left(-1\right)^{n-1}\frac{2^n+1}{n}x^n+\cdots$

Published on 27 Aug 2018 - 22:14

score 214 · Answer 6 · 2018-08-28 12:13:09

214

Views

Evaluating the sum $\sum\limits_{k=0}^{15} (-1)^k \cos^{560} (k\pi/16).$

Published on 28 Aug 2018 - 12:13

score 518 · Answer 7 · 2018-08-28 12:42:16

518

Views

Compute $1^{1/3}+2^{1/3}.........7999^{1/3}$

Published on 28 Aug 2018 - 12:42

score 134 · Answer 8 · 2026-03-25 22:04:50

134

Views

Asymptotic approximation of the expression $\sum _{i=0}^n \log (\binom{n}{i}+1)$

Published on 25 Mar 2026 - 22:04

score 721 · Answer 9 · 2026-03-25 04:44:21

721

Views

Changing finite double summation $\sum_{i=1}^n\sum_{k=1}^i f(k) = \sum_{k=1}^n\sum_{i=1}^k f(n+1-k).$

Published on 25 Mar 2026 - 4:44

score 275 · Answer 10 · 2026-03-25 03:07:40

275

Views

Constant term in Stirling type formula for $\sum^N_{n=1} H_n \cdot \ln(n)$

Published on 25 Mar 2026 - 3:07

score 51 · Answer 11 · 2026-03-25 17:24:38

51

Views

Estimate (on paper) the output of this function

Published on 25 Mar 2026 - 17:24

score 112 · Answer 12 · 2018-08-29 12:00:14

112

Views

Double Summation Multiplication

Published on 29 Aug 2018 - 12:00

score 661 · Answer 13 · 2026-03-26 07:58:36

661

Views

Find the set of points where two functions are equal

Published on 26 Mar 2026 - 7:58

score 122 · Answer 14 · 2026-03-25 22:06:16

122

Views

Asymptotic approximation of the expression $\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$

Published on 25 Mar 2026 - 22:06

score 220 · Answer 15 · 2018-08-29 17:24:22

220

Views

Finding value of a summation series.

Published on 29 Aug 2018 - 17:24

List Question

An Euler type sum: $\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n\cdot 4^n}{2n \choose n}$, where $H_n^{(2)}=\sum\limits_{k=1}^{n}\frac{1}{k^2}$

How do I convert a sum (from $1$ to $n$) of powers into a function of $n$?

Sum to n terms the series $\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$

Sum to n terms of the following series: $2 \cdot 2 + 6 \cdot 4 + 12 \cdot 8 + 20 \cdot 16 + \cdots $

Show that $\ln\left(1+3x + 2x^2\right) = 3x - \frac{5x^2}{2} + \frac{9x^3}{3} -\cdots + \left(-1\right)^{n-1}\frac{2^n+1}{n}x^n+\cdots$

Evaluating the sum $\sum\limits_{k=0}^{15} (-1)^k \cos^{560} (k\pi/16).$

Compute $1^{1/3}+2^{1/3}.........7999^{1/3}$

Asymptotic approximation of the expression $\sum _{i=0}^n \log (\binom{n}{i}+1)$

Changing finite double summation $\sum_{i=1}^n\sum_{k=1}^i f(k) = \sum_{k=1}^n\sum_{i=1}^k f(n+1-k).$

Constant term in Stirling type formula for $\sum^N_{n=1} H_n \cdot \ln(n)$

Estimate (on paper) the output of this function

Double Summation Multiplication

Find the set of points where two functions are equal

Asymptotic approximation of the expression $\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$

Finding value of a summation series.

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