Question

How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$

score 108 · Answer 1 · 2026-03-23 23:13:10

108

Views

How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$

Published on 23 Mar 2026 - 23:13

score 135 · Answer 2 · 2026-03-24 12:37:44

135

Views

Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$

Published on 24 Mar 2026 - 12:37

score 382 · Answer 3 · 2026-03-21 12:28:57

382

Views

Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?

Published on 21 Mar 2026 - 12:28

score 155 · Answer 4 · 2026-03-20 08:54:35

155

Views

Finding a closed form for a simple product

Published on 20 Mar 2026 - 8:54

score 55 · Answer 5 · 2026-03-22 08:22:18

55

Views

For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?

Published on 22 Mar 2026 - 8:22

score 99 · Answer 6 · 2026-03-24 18:45:02

99

Views

Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$

Published on 24 Mar 2026 - 18:45

score 128 · Answer 7 · 2026-03-22 22:57:09

128

Views

How can one show that $\int_{0}^{1}{x\ln{x}\ln(1-x^2)\over \sqrt{1-x^2}}\mathrm dx=4-{\pi^2\over 4}-\ln{4}?$

Published on 22 Mar 2026 - 22:57

score 171 · Answer 8 · 2026-03-20 06:29:41

171

Views

Exercises about closed form formula of recursive sequence.

Published on 20 Mar 2026 - 6:29

score 656 · Answer 9 · 2026-03-23 06:04:09

656

Views

Simplify and determine a closed form for a nested summation

Published on 23 Mar 2026 - 6:04

score 447 · Answer 10 · 2026-03-23 00:26:08

447

Views

Direction in closed form of recurrence relation

Published on 23 Mar 2026 - 0:26

score 125 · Answer 11 · 2026-03-22 19:51:06

125

Views

Does a closed form formula for the series $\sum\limits_{n = 1}^N {\frac{{1 + {x^{2n - 1}}}}{{1 + {x^{2n + 1}}}}}$ exist?

Published on 22 Mar 2026 - 19:51

score 261 · Answer 12 · 2026-03-24 20:33:37

261

Views

Showing that $\sum_{j=0}^{2n-1}{\cos^n(\frac{j\pi}{2n})(2\cos(\frac{2j\pi}n)+1)\cos(\frac{j\pi}2-\frac{2j\pi}n)}$ is never an integer for $n>10$

Published on 24 Mar 2026 - 20:33

score 56 · Answer 13 · 2026-03-24 20:33:25

56

Views

The series $\sum_{n=1}^\infty\frac{1}{k^{\lfloor{\log_m(n)}\rfloor}}$

Published on 24 Mar 2026 - 20:33

score 212 · Answer 14 · 2026-03-19 21:05:01

212

Views

Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$

Published on 19 Mar 2026 - 21:05

score 1.8k · Answer 15 · 2026-03-20 17:24:39

1.8k

Views

What is the closed form of the sequence : $ a_{n+2} = 2a_{n+1}-a_n+2^n+2 $

Published on 20 Mar 2026 - 17:24

List Question

How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$

Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$

Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?

Finding a closed form for a simple product

For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?

Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$

How can one show that $\int_{0}^{1}{x\ln{x}\ln(1-x^2)\over \sqrt{1-x^2}}\mathrm dx=4-{\pi^2\over 4}-\ln{4}?$

Exercises about closed form formula of recursive sequence.

Simplify and determine a closed form for a nested summation

Direction in closed form of recurrence relation

Does a closed form formula for the series $\sum\limits_{n = 1}^N {\frac{{1 + {x^{2n - 1}}}}{{1 + {x^{2n + 1}}}}}$ exist?

Showing that $\sum_{j=0}^{2n-1}{\cos^n(\frac{j\pi}{2n})(2\cos(\frac{2j\pi}n)+1)\cos(\frac{j\pi}2-\frac{2j\pi}n)}$ is never an integer for $n>10$

The series $\sum_{n=1}^\infty\frac{1}{k^{\lfloor{\log_m(n)}\rfloor}}$

Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$

What is the closed form of the sequence : $ a_{n+2} = 2a_{n+1}-a_n+2^n+2 $

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