Question

Modification of a logarithmic function with an infinite product equation

score 61 · Answer 1 · 2023-08-04 20:35:35

61

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Modification of a logarithmic function with an infinite product equation

Published on 04 Aug 2023 - 20:35

score 73 · Answer 2 · 2023-08-05 08:52:33

73

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given $a_{n+1}=2a_n^2-1$ ($a_1>1$) prove that infinite product $\displaystyle\prod_{k=1}^n \left(1+\dfrac{1}{a_k}\right)$ converges

Published on 05 Aug 2023 - 8:52

score 64 · Answer 3 · 2023-08-12 16:46:44

64

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Peculiar behavior of an infinite product of the function $\tan(x)$

Published on 12 Aug 2023 - 16:46

score 69 · Answer 4 · 2023-08-14 00:27:15

69

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Completing the summation equality with the example $\sum_{n=2}^{\infty} \sum_{m=1}^n \frac{\prod_{i=0}^n(i+2)}{n+m}$

Published on 14 Aug 2023 - 0:27

score 60 · Answer 5 · 2026-03-28 00:46:43

60

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Simplifying $\sin(nx)$ series notation

Published on 28 Mar 2026 - 0:46

score 45 · Answer 6 · 2023-08-21 19:56:16

45

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Binomial form of $\frac{A(A+1) \cdots (A+n-2)}{(s+A-1)(s+A-2) \cdots (s+A-i+1)}$

Published on 21 Aug 2023 - 19:56

score 63 · Answer 7 · 2023-08-25 16:21:14

63

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clarification on gamma function to Pochhammer notation

Published on 25 Aug 2023 - 16:21

score 331 · Answer 8 · 2023-09-06 19:10:51

331

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How to solve $\prod_{n=0}^\infty \left(1 - x^{2^n}\right)$?

Published on 06 Sep 2023 - 19:10

score 83 · Answer 9 · 2023-09-07 18:27:18

83

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How to evaluate the infinite product $\prod_{n\geq1}\prod_{k\geq1}e^{\frac{(-1)^{n+k}}{n^2k^2+3nk^2+3n^2k+2k^2+2n^2+9kn+6k+6n+4}}$

Published on 07 Sep 2023 - 18:27

score 107 · Answer 10 · 2023-09-08 15:29:57

107

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Fun little integral and infinite product

Published on 08 Sep 2023 - 15:29

score 13 · Answer 11 · 2026-02-28 22:16:50

Could I have computed my infinite product without using the fact that any $\prod_{a\geq b}c^{d_a}\implies c^{\sum_{a\geq b}d_a}$?

score 44 · Answer 12 · 2023-09-18 02:04:47

$a\in(0,\frac{1}{2})$, $\{x_{k}\}_{k\ge1}$ where $x_1=a$ and $2x_{k+1}=\left(\frac{x_k}{1-x_k}\right)^2$. Evaluate $\prod_{k=1}^\infty(1-x_k)$

score 71 · Answer 13 · 2023-09-26 09:07:52

71

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Maximal ideals of the infinite product (copy) of $\mathbb{F}_2$

Published on 26 Sep 2023 - 9:07

score 1k · Answer 14 · 2023-09-28 06:20:14

1k

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Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$

Published on 28 Sep 2023 - 6:20

score 152 · Answer 15 · 2023-09-29 04:46:10

152

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Given that: $1=(1+\frac{1}{3})(1-\frac{1}{5})(1-\frac{1}{7})(1+\frac{1}{9})(1+\frac{1}{11})\cdots$, Prove the following:

Published on 29 Sep 2023 - 4:46

List Question

Modification of a logarithmic function with an infinite product equation

given $a_{n+1}=2a_n^2-1$ ($a_1>1$) prove that infinite product $\displaystyle\prod_{k=1}^n \left(1+\dfrac{1}{a_k}\right)$ converges

Peculiar behavior of an infinite product of the function $\tan(x)$

Completing the summation equality with the example $\sum_{n=2}^{\infty} \sum_{m=1}^n \frac{\prod_{i=0}^n(i+2)}{n+m}$

Simplifying $\sin(nx)$ series notation

Binomial form of $\frac{A(A+1) \cdots (A+n-2)}{(s+A-1)(s+A-2) \cdots (s+A-i+1)}$

clarification on gamma function to Pochhammer notation

How to solve $\prod_{n=0}^\infty \left(1 - x^{2^n}\right)$?

How to evaluate the infinite product $\prod_{n\geq1}\prod_{k\geq1}e^{\frac{(-1)^{n+k}}{n^2k^2+3nk^2+3n^2k+2k^2+2n^2+9kn+6k+6n+4}}$

Fun little integral and infinite product

Could I have computed my infinite product without using the fact that any $\prod_{a\geq b}c^{d_a}\implies c^{\sum_{a\geq b}d_a}$?

$a\in(0,\frac{1}{2})$, $\{x_{k}\}_{k\ge1}$ where $x_1=a$ and $2x_{k+1}=\left(\frac{x_k}{1-x_k}\right)^2$. Evaluate $\prod_{k=1}^\infty(1-x_k)$

Maximal ideals of the infinite product (copy) of $\mathbb{F}_2$

Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$

Given that: $1=(1+\frac{1}{3})(1-\frac{1}{5})(1-\frac{1}{7})(1+\frac{1}{9})(1+\frac{1}{11})\cdots$, Prove the following:

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