Question

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

score 2.1k · Answer 1 · 2026-03-29 05:43:27

2.1k

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Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Published on 29 Mar 2026 - 5:43

score 6.4k · Answer 2 · 2013-03-26 07:54:36

6.4k

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Why can't $\pi$ be expressed as a fraction?

Published on 26 Mar 2013 - 7:54

score 125 · Answer 3 · 2013-03-27 14:00:46

125

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Geometric explanations of approximations of $\pi$

Published on 27 Mar 2013 - 14:00

score 123 · Answer 4 · 2026-02-24 00:23:47

123

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Is $\pi$ a "somewhat normal number"?

Published on 24 Feb 2026 - 0:23

score 313 · Answer 5 · 2026-03-07 16:53:48

313

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What are Padé approximants and how are they used for series acceleration?

Published on 07 Mar 2026 - 16:53

score 61 · Answer 6 · 2020-08-05 16:54:13

61

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Is there a positive integer $n$ such that $\lfloor 10^n\pi\rfloor$ is palindromic?

Published on 05 Aug 2020 - 16:54

score 440 · Answer 7 · 2020-08-09 04:45:53

440

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Q: Babylonian tablet for pi=3.125

Published on 09 Aug 2020 - 4:45

score 518 · Answer 8 · 2020-08-09 22:53:16

518

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Iterative algorithm for $\pi$?

Published on 09 Aug 2020 - 22:53

score 275 · Answer 9 · 2020-08-10 15:52:44

275

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How do you prove $\pi =\sqrt{12}\sum_{n\ge 0}\frac{(-1)^n}{3^n(2n+1)}$?

Published on 10 Aug 2020 - 15:52

score 195 · Answer 10 · 2020-08-15 03:06:18

195

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How to calculate $\theta = \arcsin\left(\frac{1}{2}\right)$ with given four function calculator?

Published on 15 Aug 2020 - 3:06

score 135 · Answer 11 · 2026-03-30 16:06:01

135

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Why $8^{\frac{1}{3}}$ is $1$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$

Published on 30 Mar 2026 - 16:06

score 66 · Answer 12 · 2020-08-16 06:29:21

66

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What are the differences between $r^n\text{cis}(\theta)$ and $r^n \text{cis}(\frac{2k\mathrm\pi}n)$?

Published on 16 Aug 2020 - 6:29

score 1.6k · Answer 13 · 2026-03-30 16:03:41

1.6k

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Ramanujan's $\sqrt{\frac{\pi e}{2}}$ formula

Published on 30 Mar 2026 - 16:03

score 533 · Answer 14 · 2020-08-18 14:31:46

533

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Proving that $\pi$ can be computed by colliding blocks

Published on 18 Aug 2020 - 14:31

score 137 · Answer 15 · 2020-08-22 10:51:40

137

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Why does the sum of such series involve $\pi$?

Published on 22 Aug 2020 - 10:51

List Question

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Why can't $\pi$ be expressed as a fraction?

Geometric explanations of approximations of $\pi$

Is $\pi$ a "somewhat normal number"?

What are Padé approximants and how are they used for series acceleration?

Is there a positive integer $n$ such that $\lfloor 10^n\pi\rfloor$ is palindromic?

Q: Babylonian tablet for pi=3.125

Iterative algorithm for $\pi$?

How do you prove $\pi =\sqrt{12}\sum_{n\ge 0}\frac{(-1)^n}{3^n(2n+1)}$?

How to calculate $\theta = \arcsin\left(\frac{1}{2}\right)$ with given four function calculator?

Why $8^{\frac{1}{3}}$ is $1$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$

What are the differences between $r^n\text{cis}(\theta)$ and $r^n \text{cis}(\frac{2k\mathrm\pi}n)$?

Ramanujan's $\sqrt{\frac{\pi e}{2}}$ formula

Proving that $\pi$ can be computed by colliding blocks

Why does the sum of such series involve $\pi$?

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