Ok, you say, “this works”; this is a series representation for . Ok, it is but why?
Now if you tell me: and that and term by term integration yields:
I’d remind you of: “interval of absolute convergence” and remind you that the series for does NOT converge at and that one has to be in the open interval of convergence to justify term by term integration.
True, the series DOES converge to but it is NOT that elementary to see. 🙂
(Yes, the series IS correct…but the justification is trickier than merely doing the “obvious”).