Question

Show that $\lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists an is positive

score 284 · Answer 1 · 2011-02-12 17:59:29

284

Views

Show that $\lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists an is positive

Published on 12 Feb 2011 - 17:59

score 85 · Answer 2 · 2011-06-19 17:21:29

85

Views

Is there a sequence of $v_{n}$ such that $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}> 1$?

Published on 19 Jun 2011 - 17:21

score 4.5k · Answer 3 · 2011-11-03 19:25:43

4.5k

Views

The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?

Published on 03 Nov 2011 - 19:25

score 303 · Answer 4 · 2012-01-16 11:55:27

303

Views

Limit $\lim_{n\to+\infty} (1-\frac1{2^2})(1-\frac1{3^2})\cdot \cdots \cdot(1-\frac{1}{n^2})$ and series $\sum_{n=2}^{\infty} \ln(1-\frac1{n^2})$

Published on 16 Jan 2012 - 11:55

score 164 · Answer 5 · 2026-03-25 14:34:36

164

Views

factoring infinite products of $q$-series with constant term equal to 1

Published on 25 Mar 2026 - 14:34

score 338 · Answer 6 · 2012-03-12 01:57:58

338

Views

Convergence of $\prod\limits_{n=0}^\infty ( 1+u_{n}) $ when $\sum\limits_{n=0}^{\infty }u_{n}$ and $\sum \limits_{n=0}^\infty u_{n}^{2}$ diverge

Published on 12 Mar 2012 - 1:57

score 2.8k · Answer 7 · 2012-03-12 04:29:27

2.8k

Views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Published on 12 Mar 2012 - 4:29

score 632 · Answer 8 · 2012-05-01 17:21:27

632

Views

Another evaluating limit question: $\lim\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$

Published on 01 May 2012 - 17:21

score 3.1k · Answer 9 · 2026-04-01 21:37:28

3.1k

Views

Result of the product $0.9 \times 0.99 \times 0.999 \times ...$

Published on 01 Apr 2026 - 21:37

score 15.4k · Answer 10 · 2012-06-12 13:36:36

15.4k

Views

Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$

Published on 12 Jun 2012 - 13:36

score 4.2k · Answer 11 · 2012-06-14 03:44:09

4.2k

Views

Infinite products - reference needed!

Published on 14 Jun 2012 - 3:44

score 1.4k · Answer 12 · 2012-07-11 18:49:20

1.4k

Views

Infinite product

Published on 11 Jul 2012 - 18:49

score 494 · Answer 13 · 2026-04-01 20:23:46

494

Views

Hypergeometric formulas for the Rogers-Ramanujan identities?

Published on 01 Apr 2026 - 20:23

score 519 · Answer 14 · 2012-09-20 01:26:06

519

Views

Proving that under certain conditions, the product of a sequence of holomorphic functions converges uniformly

Published on 20 Sep 2012 - 1:26

score 243 · Answer 15 · 2012-09-24 18:30:18

243

Views

an infinite system of infinite products

Published on 24 Sep 2012 - 18:30

List Question

Show that $\lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists an is positive

Is there a sequence of $v_{n}$ such that $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}> 1$?

The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?

Limit $\lim_{n\to+\infty} (1-\frac1{2^2})(1-\frac1{3^2})\cdot \cdots \cdot(1-\frac{1}{n^2})$ and series $\sum_{n=2}^{\infty} \ln(1-\frac1{n^2})$

factoring infinite products of $q$-series with constant term equal to 1

Convergence of $\prod\limits_{n=0}^\infty ( 1+u_{n}) $ when $\sum\limits_{n=0}^{\infty }u_{n}$ and $\sum \limits_{n=0}^\infty u_{n}^{2}$ diverge

Proof of a theorem of Cauchy's on the convergence of an infinite product

Another evaluating limit question: $\lim\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$

Result of the product $0.9 \times 0.99 \times 0.999 \times ...$

Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$

Infinite products - reference needed!

Infinite product

Hypergeometric formulas for the Rogers-Ramanujan identities?

Proving that under certain conditions, the product of a sequence of holomorphic functions converges uniformly

an infinite system of infinite products

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