User:Sbp/Maths

Mathematical notes related to Wikipedia articles.

Even Madhava series

The Leibniz formula for π is...

$1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \;=\;{\frac {\pi }{4}}$

Or, using summation notation:

$\sum _{n=0}^{\infty }\,{\frac {(-1)^{n}}{2n+1}}\;=\;{\frac {\pi }{4}}$

But what happens if we use even denominators?

${\frac {1}{2}}\,-\,{\frac {1}{4}}\,+\,{\frac {1}{6}}\,-\,{\frac {1}{8}}\,+\,\cdots \;=\;?$

This turns out to be:

$\sum _{n=1}^{\infty }\,{\frac {(-1)^{n+1}}{2n}}\;=\;{\frac {\ln(2)}{2}}$

Of course we can extend this to the Dirichlet beta function series too:

$\beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}}$

Noting that $\beta (1)={\frac {\pi }{4}}$ and $\beta (2)=G$ where G is Catalan's constant.

This becomes, in even denominator terms:

$\gamma (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{(2n)^{s}}}$

Where $\gamma (1)={\frac {\ln(2)}{2}}$. $\gamma (2)$ may be ${\frac {\pi ^{2}}{48}}$, and $\gamma (3)$ may be ${\frac {3\zeta (3)}{32}}$ where $\zeta (3)$ is Apéry's constant.

This is also comparable to the Dirichlet eta function:

Table of some of the results

s	$\beta (s)$	$\gamma (s)$	$\eta (s)$
1	${\frac {\pi }{4}}$	${\frac {\ln(2)}{2}}$	$\ln(2)$
2	$G$	${\frac {\pi ^{2}}{48}}?$	${\frac {\pi ^{2}}{12}}$
3	${\frac {\pi ^{3}}{32}}$	${\frac {3\zeta (3)}{32}}?$	${\frac {3\zeta (3)}{4}}$

So these values are 1/2, 1/4, 1/8 as much for $\gamma$ as for $\eta$, or apparently:

$\gamma (s)={\frac {\eta (s)}{2^{s}}}$