Question

$\prod_{n=1}(1-x^n)^6=\sum_{-\infty}^{\infty}x^{n^2}\sum_{n\geq1}(2n+1)^2x^{n(n+1)}-\sum_{-\infty}^{\infty}x^{n(n+1)}\sum_{n\geq1}(2n)^2x^{n^2}$

score 69 · Answer 1 · 2020-09-12 04:54:44

69

Views

$\prod_{n=1}(1-x^n)^6=\sum_{-\infty}^{\infty}x^{n^2}\sum_{n\geq1}(2n+1)^2x^{n(n+1)}-\sum_{-\infty}^{\infty}x^{n(n+1)}\sum_{n\geq1}(2n)^2x^{n^2}$

Published on 12 Sep 2020 - 4:54

score 224 · Answer 2 · 2026-03-29 10:07:50

224

Views

Why would $1^{-\infty}$ not be 1?

Published on 29 Mar 2026 - 10:07

score 56 · Answer 3 · 2026-03-25 14:34:33

56

Views

$q$-expansion of Klein's absolute invariant using infinite products

Published on 25 Mar 2026 - 14:34

score 159 · Answer 4 · 2020-09-16 18:56:25

159

Views

What is the advantage of using an indexing set?

Published on 16 Sep 2020 - 18:56

score 171 · Answer 5 · 2020-09-17 01:17:12

171

Views

Can anything be said about continuous functions whose domain is an infinite product of topological spaces?

Published on 17 Sep 2020 - 1:17

score 102 · Answer 6 · 2020-09-20 07:28:27

102

Views

What is this sequence for this analog of Engel products?

Published on 20 Sep 2020 - 7:28

score 40 · Answer 7 · 2026-03-30 20:44:26

40

Views

Prove the convergence of the infinite product involving nested radicals

Published on 30 Mar 2026 - 20:44

score 71 · Answer 8 · 2020-10-01 15:15:56

71

Views

If $\sum a_n, \sum a_n^2, \dots , \sum a_n^{k-1}$ and $\sum |a_n|^k$ are all convergent than $\prod (1+a_n)$ is also convergent.

Published on 01 Oct 2020 - 15:15

score 33 · Answer 9 · 2020-10-01 17:00:21

33

Views

Infinite product of cosine functions

Published on 01 Oct 2020 - 17:00

score 103 · Answer 10 · 2020-10-02 07:05:09

103

Views

Convergence of $ \prod_{n=1}^{\infty}\Bigg\{ \left(1+\frac{1}{n} \right)^x\left(1-\frac{x}{n}\right)\Bigg\}$

Published on 02 Oct 2020 - 7:05

score 112 · Answer 11 · 2026-03-27 19:08:23

112

Views

Expanding function into formal infinite product

Published on 27 Mar 2026 - 19:08

score 38 · Answer 12 · 2020-10-06 02:39:45

38

Views

$ \prod_{n=1}^{\infty}(1+x^{2^{n-1}})=\dfrac{1}{1-x}\quad (|x|<1) $

Published on 06 Oct 2020 - 2:39

score 96 · Answer 13 · 2020-10-06 16:44:24

96

Views

Does $\sin\vartheta=2^n\sin\frac{\vartheta}{2^n}\prod\limits_{k=1}^n\cos\frac{\vartheta}{2^k}$ imply $\vartheta=1$?

Published on 06 Oct 2020 - 16:44

score 85 · Answer 14 · 2020-10-12 20:51:29

85

Views

For which $x \in \mathbb C$ does the product $\prod_{n=1}^{\infty} (1+\frac{1}{n})^x(1-\frac{x}{n})$ converge?

Published on 12 Oct 2020 - 20:51

score 67 · Answer 15 · 2020-10-12 20:55:56

67

Views

If $k \in \mathbb N$ and if $\sum_n a_n, \sum_n a_n^2, \dots, \sum_n |a_n|^k$ converges then $\prod_n (1+a_n)$ converges

Published on 12 Oct 2020 - 20:55

List Question

$\prod_{n=1}(1-x^n)^6=\sum_{-\infty}^{\infty}x^{n^2}\sum_{n\geq1}(2n+1)^2x^{n(n+1)}-\sum_{-\infty}^{\infty}x^{n(n+1)}\sum_{n\geq1}(2n)^2x^{n^2}$

Why would $1^{-\infty}$ not be 1?

$q$-expansion of Klein's absolute invariant using infinite products

What is the advantage of using an indexing set?

Can anything be said about continuous functions whose domain is an infinite product of topological spaces?

What is this sequence for this analog of Engel products?

Prove the convergence of the infinite product involving nested radicals

If $\sum a_n, \sum a_n^2, \dots , \sum a_n^{k-1}$ and $\sum |a_n|^k$ are all convergent than $\prod (1+a_n)$ is also convergent.

Infinite product of cosine functions

Convergence of $ \prod_{n=1}^{\infty}\Bigg\{ \left(1+\frac{1}{n} \right)^x\left(1-\frac{x}{n}\right)\Bigg\}$

Expanding function into formal infinite product

$ \prod_{n=1}^{\infty}(1+x^{2^{n-1}})=\dfrac{1}{1-x}\quad (|x|<1) $

Does $\sin\vartheta=2^n\sin\frac{\vartheta}{2^n}\prod\limits_{k=1}^n\cos\frac{\vartheta}{2^k}$ imply $\vartheta=1$?

For which $x \in \mathbb C$ does the product $\prod_{n=1}^{\infty} (1+\frac{1}{n})^x(1-\frac{x}{n})$ converge?

If $k \in \mathbb N$ and if $\sum_n a_n, \sum_n a_n^2, \dots, \sum_n |a_n|^k$ converges then $\prod_n (1+a_n)$ converges

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