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Cauchy Product of two simple geometric series

score 1k · Answer 1 · 2015-12-16 22:57:12

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Cauchy Product of two simple geometric series

Published on 16 Dec 2015 - 22:57

score 1.4k · Answer 2 · 2015-12-24 13:38:55

1.4k

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Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product.

Published on 24 Dec 2015 - 13:38

score 58 · Answer 3 · 2016-01-27 18:33:34

58

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Can someone expain to me what's going on (binomial coefficient)?

Published on 27 Jan 2016 - 18:33

score 87 · Answer 4 · 2016-02-06 20:40:51

87

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Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Published on 06 Feb 2016 - 20:40

score 38 · Answer 5 · 2016-02-14 12:40:06

38

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How to prove convergence for the following series

Published on 14 Feb 2016 - 12:40

score 531 · Answer 6 · 2016-04-16 20:29:24

531

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Applying Cauchy product to formal power series

Published on 16 Apr 2016 - 20:29

score 271 · Answer 7 · 2016-04-29 08:49:59

271

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For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

Published on 29 Apr 2016 - 8:49

score 1.5k · Answer 8 · 2016-04-29 17:15:05

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Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

Published on 29 Apr 2016 - 17:15

score 142 · Answer 9 · 2016-05-05 14:36:50

142

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The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

Published on 05 May 2016 - 14:36

score 58 · Answer 10 · 2016-05-11 21:36:19

58

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Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Published on 11 May 2016 - 21:36

score 174 · Answer 11 · 2016-05-25 17:22:16

174

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Laurent series of $\frac{e^z}{z^2+1}$

Published on 25 May 2016 - 17:22

score 616 · Answer 12 · 2016-06-12 17:47:41

616

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$A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

Published on 12 Jun 2016 - 17:47

score 976 · Answer 13 · 2017-05-09 01:45:21

976

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Suppose $\sum a_n$ converges absolutely and $\sum b_n$ converges. Give an example where the Cauchy product does not converge absolutely.

Published on 09 May 2017 - 1:45

score 1.6k · Answer 14 · 2017-05-23 20:34:57

1.6k

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Convergent Cauchy product of divergent series

Published on 23 May 2017 - 20:34

score 141 · Answer 15 · 2017-06-21 23:41:32

141

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The series $\sum_{k=1}^\infty\sum_{n=1}^k k^{-3}/(2n-1)$

Published on 21 Jun 2017 - 23:41

List Question

Cauchy Product of two simple geometric series

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product.

Can someone expain to me what's going on (binomial coefficient)?

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

How to prove convergence for the following series

Applying Cauchy product to formal power series

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Laurent series of $\frac{e^z}{z^2+1}$

$A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

Suppose $\sum a_n$ converges absolutely and $\sum b_n$ converges. Give an example where the Cauchy product does not converge absolutely.

Convergent Cauchy product of divergent series

The series $\sum_{k=1}^\infty\sum_{n=1}^k k^{-3}/(2n-1)$

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