Result of Infinite Sum of Rational Functions

Question

I am trying to understand how this infinite sum is calculated: $$\sum_{k=0}^{\infty}\frac{1}{4k+2}-\frac{1}{4k+4}$$

I tried some common techniques but I can't find a way to prove what this converges into, which is $\frac{ln(2)}{2}$. I tried converting into an integral to calculate but I didn't get far, same with Fourier series. Is there a known way to solve this?

If you multiply with $2$ , you should see the pattern of the sum easier and why the resulting sum converges to $\ln(2)$. — Peter, Oct 10 '23 at 16:38

score 1 · Answer 1 · answered Oct 10 '23 at 16:49

Hint If we rewrite the series as $$\frac{1}{2} \sum_{k = 0}^\infty \left(\frac{1}{2 k + 1} - \frac{1}{2 k + 2}\right),$$ we can see that expanding gives $$\frac{1}{2}\left(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots\right) .$$ Can you identify the series here as the value of a well-known Taylor series at some point?

Result of Infinite Sum of Rational Functions

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