Question

How to prove $\lim_{x\rightarrow0^+}\sum_{k=1}^{+\infty}\frac{\sin{(x\sqrt{k})}}{k}=\pi$?

score 247 · Answer 1 · 2026-04-12 22:07:04

247

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How to prove $\lim_{x\rightarrow0^+}\sum_{k=1}^{+\infty}\frac{\sin{(x\sqrt{k})}}{k}=\pi$?

Published on 12 Apr 2026 - 22:07

score 878 · Answer 2 · 2026-04-12 19:32:54

878

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Proving whether a series converges (Wolfram Alpha says it diverges)

Published on 12 Apr 2026 - 19:32

score 76 · Answer 3 · 2026-04-12 09:33:23

76

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Limit of series of sine terms with random arguments

Published on 12 Apr 2026 - 9:33

score 109 · Answer 4 · 2026-04-09 03:40:59

109

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Do the second differences of the fifth powers count the sphere packing of a polyhedron?

Published on 09 Apr 2026 - 3:40

score 225 · Answer 5 · 2026-04-12 19:09:41

225

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Taylor series: criteria for maximum error attained at boundary?

Published on 12 Apr 2026 - 19:09

score 122 · Answer 6 · 2018-02-08 18:15:50

122

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$\lim\limits_{n\rightarrow \infty }\sum\limits_{k=0}^{n}\frac{(-1)^{k}\binom{n}{k}}{2k+1}=0$

Published on 08 Feb 2018 - 18:15

score 108 · Answer 7 · 2026-04-12 09:00:49

108

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General formula for Evaluating $\sum_{n=0}^\infty n^ar^n$ where $ |r|<1 , a\ge0$

Published on 12 Apr 2026 - 9:00

score 235 · Answer 8 · 2026-04-09 00:54:34

235

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Upper bound on a series: $\sum_{k = M}^{\infty} (a/\sqrt{k})^k$

Published on 09 Apr 2026 - 0:54

score 123 · Answer 9 · 2026-03-29 05:43:39

123

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Convergence of the series $\sum_{n=1}^{\infty}\frac{1}{4^{\omega(n)}n}$

Published on 29 Mar 2026 - 5:43

score 713 · Answer 10 · 2026-04-11 23:55:53

713

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Consider the following sequence $1234567891011121314 . . . 9999899999100000$, how many times the block "2016" appears?

Published on 11 Apr 2026 - 23:55

score 36 · Answer 11 · 2018-02-09 02:02:11

If $f(x+1)=\frac12(f(x)+f(x-2))$ for every $x$ then $f(x)$ converges when $x\to\infty$? And what would be the limit?

score 34 · Answer 12 · 2026-03-25 06:04:29

34

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Expected value of an r.v. that counts the number of pickable values that never appear in a random sequence

Published on 25 Mar 2026 - 6:04

score 112 · Answer 13 · 2026-03-25 18:44:12

112

Views

Sequences unbounded above diverges?

Published on 25 Mar 2026 - 18:44

score 689 · Answer 14 · 2026-04-12 18:30:27

689

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20 cards facing down

Published on 12 Apr 2026 - 18:30

score 197 · Answer 15 · 2026-04-07 04:46:34

197

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Pointwise and uniform convergence of $\sum_{n=0}^\infty \frac{x^n}{e^{nx}}, n \in \mathbb{N}$

Published on 07 Apr 2026 - 4:46

List Question

How to prove $\lim_{x\rightarrow0^+}\sum_{k=1}^{+\infty}\frac{\sin{(x\sqrt{k})}}{k}=\pi$?

Proving whether a series converges (Wolfram Alpha says it diverges)

Limit of series of sine terms with random arguments

Do the second differences of the fifth powers count the sphere packing of a polyhedron?

Taylor series: criteria for maximum error attained at boundary?

$\lim\limits_{n\rightarrow \infty }\sum\limits_{k=0}^{n}\frac{(-1)^{k}\binom{n}{k}}{2k+1}=0$

General formula for Evaluating $\sum_{n=0}^\infty n^ar^n$ where $ |r|<1 , a\ge0$

Upper bound on a series: $\sum_{k = M}^{\infty} (a/\sqrt{k})^k$

Convergence of the series $\sum_{n=1}^{\infty}\frac{1}{4^{\omega(n)}n}$

Consider the following sequence $1234567891011121314 . . . 9999899999100000$, how many times the block "2016" appears?

If $f(x+1)=\frac12(f(x)+f(x-2))$ for every $x$ then $f(x)$ converges when $x\to\infty$? And what would be the limit?

Expected value of an r.v. that counts the number of pickable values that never appear in a random sequence

Sequences unbounded above diverges?

20 cards facing down

Pointwise and uniform convergence of $\sum_{n=0}^\infty \frac{x^n}{e^{nx}}, n \in \mathbb{N}$

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