Wikipedia:Reference desk/Archives/Mathematics/2013 May 11

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May 11

Ramanujan's Pi Formula

$\pi ={\sqrt {10\ -\ \sum _{n=1}^{\infty }{\left({\tfrac {1}{n}}-{\tfrac {1}{n+1}}\right)^{3}}}}$

Bruce C. Berndt, Ramanujan’s Notebooks, Part 1, Chapter 9, page 294.

Is there some easy way to prove this ? And can we include it here on Wiki, or is it copyrighted ? — 79.113.213.202 (talk) 07:31, 11 May 2013 (UTC)

Mathematics can't be copyright or patented, only applications, though see illegal number! In general what's needed for something new to be stuck into Wikipedia is that somebody else has remarked on it in a reliable source, i.e. citation needed and not just the original person who wrote it. Wikipedia is not just an indiscriminate collection of information, it has to be stuff people have shown notice in. The original person is good for a definitive account but somebody else is needed to show it is notable. Dmcq (talk) 10:22, 11 May 2013 (UTC)

I guess when I see such a pretty and elegant thing like this formula, I get swept away by its beauty, and forget all about `relevance` and `objectivity`... :-) — 79.113.235.252 (talk) 20:41, 11 May 2013 (UTC)

To prove this formula, first use partial fractions to expand the term being summed:

$\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)^{3}={\frac {1}{n^{3}(n+1)^{3}}}=\left({\frac {1}{n^{3}}}-{\frac {1}{(n+1)^{3}}}\right)-3\left({\frac {1}{n^{2}}}+{\frac {1}{(n+1)^{2}}}\right)+6\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)$

When you sum this, the first and third components telescope to give 1 and 6 respectively and the middle component becomes:

$\sum _{1}^{\infty }\left({\frac {1}{n^{2}}}+{\frac {1}{(n+1)^{2}}}\right)=2\left(\sum _{1}^{\infty }{\frac {1}{n^{2}}}\right)-1={\frac {\pi ^{2}}{3}}-1$

-see Basel problem. So:

$\sum _{1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)^{3}=1-3\left({\frac {\pi ^{2}}{3}}-1\right)+6=10-\pi ^{2}$

Gandalf61 (talk) 11:48, 11 May 2013 (UTC)

Thanks! :-) (I was thinking earlier today about using induction, and the fact that the sum of cubes is divisible through the sum itself, also for the purpose of telescoping the series). — 79.113.235.252 (talk) 20:41, 11 May 2013 (UTC)

Resolved

use of summation in fractions

Can anyone tell me how to solve (sigma)k/(k^4)+1/4 — Preceding unsigned comment added by 223.176.143.131 (talk) 14:44, 11 May 2013 (UTC)

Do you mean $\sum _{k=0}^{\infty }{\frac {k}{k^{4}+{\tfrac {1}{4}}}}\ =\ \sum _{k=0}^{\infty }{\frac {4k}{4k^{4}+1}}\ =\ 1$ ? — 79.113.235.252 (talk) 20:50, 11 May 2013 (UTC)

Normally the result of this sort of thing is fairly ghastly but in this case you could try summing the first few terms in the series and make a guess. If you want to prove your guess note the nice factorization $4k^{4}+1=(2k^{2}-2k+1)(2k^{2}+2k+1)$ and also that $2(k+1)^{2}-2(k+1)+1=2k^{2}+2k+1$. Try splitting each term in two. Dmcq (talk) 07:17, 12 May 2013 (UTC)

Didn't you rather mean

$k^{2}+(k\pm 1)^{2}=2k^{2}\pm 2k+1$

which then allows us to reduce or telescope the terms of the series, which then becomes:

$\sum _{k=1}^{\infty }{\left({\frac {1}{k^{2}+(k-1)^{2}}}-{\frac {1}{k^{2}+(k+1)^{2}}}\right)}=$

$\$

$=\ \ ({\tfrac {1}{1^{2}+0^{2}}}\underbrace {-{\tfrac {1}{1^{2}+2^{2}}})\ +\ ({\tfrac {1}{2^{2}+1^{2}}}} _{0}\underbrace {-{\tfrac {1}{2^{2}+3^{2}}})\ +\ ({\tfrac {1}{3^{2}+2^{2}}}} _{0}\underbrace {-{\tfrac {1}{3^{2}+4^{2}}})\ +\ ...} _{0}=1.$

— 79.113.216.179 (talk) 08:30, 12 May 2013 (UTC)

What I was emphasizing was that one term was the k+1 version of the other term. Dmcq (talk) 12:50, 12 May 2013 (UTC)