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Even Madhava series
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The Leibniz formula for π is...
Or, using summation notation:
But what happens if we use even denominators?
This turns out to be:
Of course we can extend this to the Dirichlet beta function series too:
Noting that and where G is Catalan's constant.
This becomes, in even denominator terms:
Where . may be , and may be where is Apéry's constant.
This is also comparable to the Dirichlet eta function:
Table of some of the results
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So these values are 1/2, 1/4, 1/8 as much for as for , or apparently: