Question

Why does this double infinite sum $\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}$ converge to $e$?

score 1.1k · Answer 1 · 2016-01-13 23:28:03

1.1k

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Why does this double infinite sum $\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}$ converge to $e$?

Published on 13 Jan 2016 - 23:28

score 165 · Answer 2 · 2016-01-15 14:49:48

165

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Asymptotic expansion question

Published on 15 Jan 2016 - 14:49

score 1.4k · Answer 3 · 2016-01-15 17:22:31

1.4k

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Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Published on 15 Jan 2016 - 17:22

score 649 · Answer 4 · 2016-01-16 04:00:24

649

Views

Show that the gamma function converges

Published on 16 Jan 2016 - 4:00

score 642 · Answer 5 · 2016-01-16 22:44:52

642

Views

Gamma function and sine

Published on 16 Jan 2016 - 22:44

score 275 · Answer 6 · 2026-03-27 14:58:42

275

Views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

Published on 27 Mar 2026 - 14:58

score 117 · Answer 7 · 2016-01-17 18:32:14

117

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How could I solve $\int_{-\infty}^{+\infty} x^2e^{-x^2}dx$ apply special function gamma

Published on 17 Jan 2016 - 18:32

score 108 · Answer 8 · 2016-01-17 18:52:20

108

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(Ahlfors, p198) Why is it clear we can write $G(z-1)=ze^{\gamma(z)}G(z)$ when deriving the Gamma function?

Published on 17 Jan 2016 - 18:52

score 97 · Answer 9 · 2026-03-27 23:57:39

97

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Same values for Gamma Function

Published on 27 Mar 2026 - 23:57

score 113 · Answer 10 · 2016-01-20 23:22:26

113

Views

If $f_1(k)$ and $f_2(k)$ reach their max. at $k_{m1}$ and $k_{m2}$, resp., show that $k_{m1} > k_{m2}$ in the following case

Published on 20 Jan 2016 - 23:22

score 191 · Answer 11 · 2016-01-21 20:43:04

191

Views

Integral involving the gamma function

Published on 21 Jan 2016 - 20:43

score 1.3k · Answer 12 · 2016-01-24 00:52:14

1.3k

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What approximations for the Gamma function's inverse appear to work 'best'?

Published on 24 Jan 2016 - 0:52

score 127 · Answer 13 · 2016-01-24 17:12:37

127

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Showing that $\int_{\mathbb{R}} e^{x(t+1)} \left(1 + \frac{e^x}{k} \right)^{-(k+1)}dx = k^{t+1} \frac{\Gamma(k-t)\Gamma(1+t)}{\Gamma{(k+1)}}$

Published on 24 Jan 2016 - 17:12

score 304 · Answer 14 · 2026-03-27 14:58:34

304

Views

Another beta integral due to Cauchy.

Published on 27 Mar 2026 - 14:58

score 648 · Answer 15 · 2016-01-29 22:20:52

648

Views

integral include lower incomplete gamma

Published on 29 Jan 2016 - 22:20

List Question

Why does this double infinite sum $\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}$ converge to $e$?

Asymptotic expansion question

Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Show that the gamma function converges

Gamma function and sine

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could I solve $\int_{-\infty}^{+\infty} x^2e^{-x^2}dx$ apply special function gamma

(Ahlfors, p198) Why is it clear we can write $G(z-1)=ze^{\gamma(z)}G(z)$ when deriving the Gamma function?

Same values for Gamma Function

If $f_1(k)$ and $f_2(k)$ reach their max. at $k_{m1}$ and $k_{m2}$, resp., show that $k_{m1} > k_{m2}$ in the following case

Integral involving the gamma function

What approximations for the Gamma function's inverse appear to work 'best'?

Showing that $\int_{\mathbb{R}} e^{x(t+1)} \left(1 + \frac{e^x}{k} \right)^{-(k+1)}dx = k^{t+1} \frac{\Gamma(k-t)\Gamma(1+t)}{\Gamma{(k+1)}}$

Another beta integral due to Cauchy.

integral include lower incomplete gamma

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