You could look at the telescopic series
Edit: Sorry wrong series
is there a way of ad hoc summing: .
without getting into the theory of Polygamma functions?
The general term is: .
Partial fractions: .
We find that: .
Crank out several terms:
. . . . . . .
. . . . . . . . .
We find that nearly all the terms cancel out.
And we are left with: .
An alternative. Let:
the sum of the given series. Using Soroban's decomposition:
Now we use the well known property:
we inmediately obtain:
The sum of the given series is:
Edited: What a strange thread! . I used Soroban's series instead of OP's series.