A series is called "divergent" if and only if it has no value!
What capea wrote is 'almost true'... in the sense I try to explain now...
The so called 'Dirichlet eta function' is defined as...
(1)
Writing the (1) as...
(2)
... You can, with a little of patience, demonstrate that (2) converges for and that is...
(3)
Now if You compute 'directly' the derivative of (1) You obtain...
(4)
... so that setting in (4) and considering (3) You obtain...
(5)
Kind regards