Question

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

score 320 · Answer 1 · 2015-02-10 22:01:22

320

Views

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

Published on 10 Feb 2015 - 22:01

score 74 · Answer 2 · 2015-02-12 16:36:11

74

Views

How to prove that if $\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$ then, for any $0<c_k<b_k$, $\lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l$?

Published on 12 Feb 2015 - 16:36

score 146 · Answer 3 · 2015-02-13 21:53:00

146

Views

The limit of product $\prod_{x=2}^k(1-x^{-2})$ as $k\to\infty$

Published on 13 Feb 2015 - 21:53

score 122 · Answer 4 · 2015-02-16 14:56:52

122

Views

Non vanishing of an infinite product

Published on 16 Feb 2015 - 14:56

score 865 · Answer 5 · 2015-02-16 22:37:43

865

Views

Order of an entire function represented as an infinite product

Published on 16 Feb 2015 - 22:37

score 94 · Answer 6 · 2015-02-28 18:05:40

94

Views

What is the value of this Infinite Product of prime numbers expression?

Published on 28 Feb 2015 - 18:05

score 178 · Answer 7 · 2015-02-28 23:19:36

178

Views

What's a "Basis of Measurable Sets?"

Published on 28 Feb 2015 - 23:19

score 454 · Answer 8 · 2026-03-26 14:25:21

454

Views

Products of torsion groups

Published on 26 Mar 2026 - 14:25

score 288 · Answer 9 · 2015-03-02 02:41:49

288

Views

$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$

Published on 02 Mar 2015 - 2:41

score 3.9k · Answer 10 · 2015-03-04 04:44:17

3.9k

Views

Is it possible to turn infinite sums into infinite products?

Published on 04 Mar 2015 - 4:44

score 169 · Answer 11 · 2015-03-07 17:19:46

169

Views

Question concerning the conditional convergence of some infinite products

Published on 07 Mar 2015 - 17:19

score 153 · Answer 12 · 2015-03-08 06:10:43

153

Views

Question concerning the influence of interchanging factors of an infinite product on the value of the product

Published on 08 Mar 2015 - 6:10

score 308 · Answer 13 · 2015-03-09 07:42:47

308

Views

How to Show $\prod_{n=1}^\infty\bigg(\frac{z^n}{n!}+e^{\frac{z}{2^n}}\bigg)$ Converges Uniformly on Compact Sets

Published on 09 Mar 2015 - 7:42

score 266 · Answer 14 · 2026-03-28 22:08:20

266

Views

Proof of Ramanujan's Identities of Euler's Function

Published on 28 Mar 2026 - 22:08

score 1.5k · Answer 15 · 2015-03-10 05:14:59

1.5k

Views

Compute the infinite product $\prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right)$

Published on 10 Mar 2015 - 5:14

List Question

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

How to prove that if $\lim_{n\to\infty} a_n \prod_{k=1}^n b_k = l>0$ then, for any $0<c_k<b_k$, $\lim_{n\to\infty} a_n \prod_{k=1}^n (b_k-c_k) < l$?

The limit of product $\prod_{x=2}^k(1-x^{-2})$ as $k\to\infty$

Non vanishing of an infinite product

Order of an entire function represented as an infinite product

What is the value of this Infinite Product of prime numbers expression?

What's a "Basis of Measurable Sets?"

Products of torsion groups

$\text{Prove }\prod_{i=1}^\infty(1+a_i) \text{ converges } \iff \sum_{n=1}^\infty a_n \text{ converges}$

Is it possible to turn infinite sums into infinite products?

Question concerning the conditional convergence of some infinite products

Question concerning the influence of interchanging factors of an infinite product on the value of the product

How to Show $\prod_{n=1}^\infty\bigg(\frac{z^n}{n!}+e^{\frac{z}{2^n}}\bigg)$ Converges Uniformly on Compact Sets

Proof of Ramanujan's Identities of Euler's Function

Compute the infinite product $\prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right)$

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