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Let denote the zeta function. Prove that
Evaluate the series
One can argue that the same technique used to evaluate the sum here can be used here as well. Unfortunately, this is not the case as the sum does not telescope. However the technique used here is the way to go.
First of all we note that
However, it is known that
Let . We note that . Hence,
which is now a matter of calculations. The sum is equal to
Evaluate the sum
First of all note that:
Hence letting we have that
The key ingredient is the observation . Then we note that
Using the arg’s property we get that
Hence the initial sum telescopes;
since which explains why pops up.