The sum should be .
I haven't thought about this deeply or anything, but this almost seems like it can be done using some kind of summation by parts argument.
Summation by parts - Wikipedia, the free encyclopedia
Let's start with the so called 'Dirichlet eta function'...
(1)
... where is the so called 'Riemann zeta function'. The goal is to find a series expansion 'somewehre around' of its derivative...
(2)
At this scope we use the [well known] Laurent expansion...
(3)
... where is the Euler's constant and the are the so called 'Stieltjes constants'. Deriving (3) we obtain...
(4)
... and is...
(5)
... so that we have all the terms we need to compute (2). We are interested to the value of (2) in , so that we stop the computation at the constant term obtaining...
(6)
... and finally...
(7)
Kind regards