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:

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


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