Question

Infinite Product Series

score 99 · Answer 1 · 2020-03-09 17:09:56

99

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Infinite Product Series

Published on 09 Mar 2020 - 17:09

score 92 · Answer 2 · 2020-03-11 04:47:58

92

Views

On proving $\Gamma (z+1)=4^z \prod _{k=1}^{\infty } \frac{\Gamma \left(\frac{z}{2^k}+\frac{1}{2}\right)}{\sqrt{\pi }}$

Published on 11 Mar 2020 - 4:47

score 75 · Answer 3 · 2020-03-12 18:46:35

75

Views

If $\;x\in\left[1,\infty \right)$, then $ x=\prod_{r=0}^\infty\left(1+\left(\frac{x-1}{x}\right)^{2^r}\right)$

Published on 12 Mar 2020 - 18:46

score 127 · Answer 4 · 2020-03-14 09:47:27

127

Views

Simplifying this expression of gamma functions

Published on 14 Mar 2020 - 9:47

score 108 · Answer 5 · 2020-03-15 03:30:50

108

Views

Study of an infinite product

Published on 15 Mar 2020 - 3:30

score 67 · Answer 6 · 2020-03-15 06:49:09

Prove of: $\prod_{n=0}^{\infty}\frac{(3n+2)^5(3n+4)^{10}(3n+6)}{(3n+1)(3n+3)^{10}(3n+5)^5}=\frac{6^5}{\Gamma^9\left(\frac{1}{3}\right)}$

score 848 · Answer 7 · 2020-03-15 07:25:26

848

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Convergence of the infinite product $\prod _{n=1} ^\infty (1+x^n)$ for $0<x<1$

Published on 15 Mar 2020 - 7:25

score 94 · Answer 8 · 2020-03-15 14:56:47

94

Views

Find the infinite product

Published on 15 Mar 2020 - 14:56

score 97 · Answer 9 · 2020-03-19 06:31:21

97

Views

Question about taking logarithms of Infinite Products and then differentiating.

Published on 19 Mar 2020 - 6:31

score 262 · Answer 10 · 2026-03-28 02:44:51

262

Views

How can I prove that $b = \prod_{i=1}^n (3 + \frac{1}{x_i})$ results not in an even, natural number, except for $n=1$ .and $x1 = 1$

Published on 28 Mar 2026 - 2:44

score 156 · Answer 11 · 2026-03-25 21:51:09

156

Views

Is the sum $\sum_{i=1}^{n}\frac{2^{\alpha_1+\ldots+\alpha_i}}{3^iv_1+\sum_{j=1}^{i}3^{j-1}2^{\alpha_1+\ldots+\alpha_n-\sum_{l>i-j}\alpha_l}}$ limited?

Published on 25 Mar 2026 - 21:51

score 203 · Answer 12 · 2026-03-26 18:58:54

203

Views

Relation between lemniscate constant and hyperbolic tangent.

Published on 26 Mar 2026 - 18:58

score 91 · Answer 13 · 2020-03-23 18:52:05

91

Views

Show that: $\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$

Published on 23 Mar 2020 - 18:52

score 50 · Answer 14 · 2020-03-24 00:54:43

50

Views

Is there a closed form of $\prod_{p \in \mathbb{P}} p^{p^{-k}}$ ,$k >1$?

Published on 24 Mar 2020 - 0:54

score 187 · Answer 15 · 2026-03-27 10:47:48

187

Views

Find the limit of an infinite product related to the alternating harmonic convergence.

Published on 27 Mar 2026 - 10:47

List Question

Infinite Product Series

On proving $\Gamma (z+1)=4^z \prod _{k=1}^{\infty } \frac{\Gamma \left(\frac{z}{2^k}+\frac{1}{2}\right)}{\sqrt{\pi }}$

If $\;x\in\left[1,\infty \right)$, then $ x=\prod_{r=0}^\infty\left(1+\left(\frac{x-1}{x}\right)^{2^r}\right)$

Simplifying this expression of gamma functions

Study of an infinite product

Prove of: $\prod_{n=0}^{\infty}\frac{(3n+2)^5(3n+4)^{10}(3n+6)}{(3n+1)(3n+3)^{10}(3n+5)^5}=\frac{6^5}{\Gamma^9\left(\frac{1}{3}\right)}$

Convergence of the infinite product $\prod _{n=1} ^\infty (1+x^n)$ for $0<x<1$

Find the infinite product

Question about taking logarithms of Infinite Products and then differentiating.

How can I prove that $b = \prod_{i=1}^n (3 + \frac{1}{x_i})$ results not in an even, natural number, except for $n=1$ .and $x1 = 1$

Is the sum $\sum_{i=1}^{n}\frac{2^{\alpha_1+\ldots+\alpha_i}}{3^iv_1+\sum_{j=1}^{i}3^{j-1}2^{\alpha_1+\ldots+\alpha_n-\sum_{l>i-j}\alpha_l}}$ limited?

Relation between lemniscate constant and hyperbolic tangent.

Show that: $\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$

Is there a closed form of $\prod_{p \in \mathbb{P}} p^{p^{-k}}$ ,$k >1$?

Find the limit of an infinite product related to the alternating harmonic convergence.

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