>>10398874Well this argument does not work unless there is a second derivative that is continuous everywhere and an assumption on all values of f'', ensuring that the function is indeed increasing globally.
Otherwise, you can just copy and paste a pattern that is twice differentiable with positive derivative at integers like this .
This is twice differentiable at integers with constant second derivative 2 (so that the third assumption is met interpreted as a limit in integers) but f(n) = 1 for each integer n, hence the series diverges.
Also, I am being lazy but this counterexample can be made continuous and even differentiable everywhere, so this is not the problem.
