Sums of the form where , sometimes are called Euler sums. There are several ways to evaluate these sums. Aabout 240 years ago, Leonhard Euler(1707-1783) in 1775 proved that for ,
In the following you can see an elementary proof of this formula in the three steps:
Now by recent formula that is also an alternate definition of the Harmonic numbers, we can write:
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 ,