By using the Flajolet-Vardi theorem we can find the value of the another amazing convergent series. Indeed, following series
I first recall the Flajolet-Vardi theorem that you can find it’s proof in my second post:
If and converges then,
This theorem shows that , because if we let , then and by this theorem
Therefore now we must find the value of . we use the Taylor expansion of and the fact that the sum is if divides and otherwise.
There are some ‘s of complex numbers. Those numbers have always non-negative real part, for the Argument we take the angle between and , so that it fits with the power series for .
We thus have
We have to take the limit . The term disappears, so we get
For we have
and also for ,