Well, most of you would have definitely proved that the alternating sum of reciprocal of odd numbers equals .
Here's a more generalized case of the series
It can be easily proved that
by making the substitution and then the integral converts to Beta Function and the final result follows by using Euler's Reflection Formula
Now, all that remains is to split the integral as
The first term evaluates to
Similarly, the second term can be evaluated by making the following substitution :
And thus the result follows.
Interesting part is that, the base case when , you get the above mentioned well known series of alternating sum of reciprocal of odd numbers that is
And I am sure, that you all must have got some other, perhaps more elegant, proof for the same. So, please do share it with all of us.