Divergent sums and substraction

Question

I have been thinking on this fact. The harmonic sum is divergent, but if I take the sum with N terms and subtract the $ln(N)$, the limit is finite when N goes to infinity ($\gamma$).

How can this be generalized to other series? Is there a formal theory?

Thanks!

Answer 1

"Cutting off" a divergent series at some point, estimating the "partial sum", then subtracting off the "divergent part" to obtain a finite limit is indeed one way of "summing" (regularizing) divergent series. We of course need to define what we mean by the terms in quotes, though. Terry Tao wrote a great blog post about this specific topic.