Question

Find $C$ such that $\frac{1}{n}\prod_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ converges to a positive number.

score 112 · Answer 1 · 2026-04-03 16:29:16

112

Views

Find $C$ such that $\frac{1}{n}\prod_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ converges to a positive number.

Published on 03 Apr 2026 - 16:29

score 159 · Answer 2 · 2026-04-03 16:29:26

159

Views

Show that $\lim_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$.

Published on 03 Apr 2026 - 16:29

score 544 · Answer 3 · 2026-03-27 05:41:55

544

Views

Curiosities of the function $Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$ where $P_n(x)$ is a sequence of all polynomials with unit coefficients

Published on 27 Mar 2026 - 5:41

score 106 · Answer 4 · 2026-04-12 21:45:07

106

Views

Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?

Published on 12 Apr 2026 - 21:45

score 221 · Answer 5 · 2026-03-25 20:42:27

221

Views

Evaluate $\sum_{n\geq 0} \mbox{arccot}(n^2 + n + 1)$

Published on 25 Mar 2026 - 20:42

score 241 · Answer 6 · 2026-04-13 03:55:51

241

Views

On an infinite harmonic series.

Published on 13 Apr 2026 - 3:55

score 110 · Answer 7 · 2022-11-04 18:33:47

110

Views

Find a closed form solution for the recurrence relation $a_n = a_{n-1} a_{n-2}$

Published on 04 Nov 2022 - 18:33

score 49 · Answer 8 · 2026-04-13 15:09:59

49

Views

Closed form for definite integral giving logit probability of first and second choices in the same nest?

Published on 13 Apr 2026 - 15:09

score 192 · Answer 9 · 2026-04-13 23:34:12

192

Views

What is the product of the areas of every regular polygon inscribed in a circle of area $1$?

Published on 13 Apr 2026 - 23:34

score 202 · Answer 10 · 2026-03-28 05:23:10

202

Views

$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.

Published on 28 Mar 2026 - 5:23

score 194 · Answer 11 · 2026-04-13 16:05:15

194

Views

Weierstrass Zeta $\zeta(x;a,b)$ inverse

Published on 13 Apr 2026 - 16:05

score 176 · Answer 12 · 2026-04-13 00:53:40

176

Views

How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

Published on 13 Apr 2026 - 0:53

score 34 · Answer 13 · 2026-04-14 03:05:45

34

Views

Closed-form value of a product $\prod_{i = 2}^n (1 + \frac{p}{i})$ with $0 \leq p \leq 2$

Published on 14 Apr 2026 - 3:05

score 70 · Answer 14 · 2022-11-29 04:39:02

70

Views

Finding closed form of recurrence relation that has a modulus term: $a_{n+1} = \frac 3 2 a_n + \big(\frac 1 2\big)^{a_n \bmod 2}$

Published on 29 Nov 2022 - 4:39

score 257 · Answer 15 · 2022-12-01 11:09:03

257

Views

Closed form for $\int \sqrt[n]{\tan x}\ dx$

Published on 01 Dec 2022 - 11:09

List Question

Find $C$ such that $\frac{1}{n}\prod_{k=1}^{n}C\left(\cos{\frac{k\pi}{2(n+1)}}+\sin{\frac{k\pi}{2(n+1)}}-1\right)$ converges to a positive number.

Show that $\lim_{n\to\infty}n\left(n\ln{n}+\ln{\sqrt{2}}-n-\sum_{k=1}^n \ln{\left(k-\frac{1}{2}\right)}\right)=\frac{1}{24}$.

Curiosities of the function $Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$ where $P_n(x)$ is a sequence of all polynomials with unit coefficients

Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?

Evaluate $\sum_{n\geq 0} \mbox{arccot}(n^2 + n + 1)$

On an infinite harmonic series.

Find a closed form solution for the recurrence relation $a_n = a_{n-1} a_{n-2}$

Closed form for definite integral giving logit probability of first and second choices in the same nest?

What is the product of the areas of every regular polygon inscribed in a circle of area $1$?

$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.

Weierstrass Zeta $\zeta(x;a,b)$ inverse

How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?

Closed-form value of a product $\prod_{i = 2}^n (1 + \frac{p}{i})$ with $0 \leq p \leq 2$

Finding closed form of recurrence relation that has a modulus term: $a_{n+1} = \frac 3 2 a_n + \big(\frac 1 2\big)^{a_n \bmod 2}$

Closed form for $\int \sqrt[n]{\tan x}\ dx$

Trending Questions

Popular # Hahtags

Popular Questions