User:Sbp/Maths

Mathematical notes related to Wikipedia articles.

Even Madhava series

The Leibniz formula for π is...

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

Or, using summation notation:

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

But what happens if we use even denominators?

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

This turns out to be:

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

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

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

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

This becomes, in even denominator terms:

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

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

This is also comparable to the Dirichlet eta function:

Table of some of the results

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

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

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