Use Abel's test to show that the complex power series converges on the unit circle (and therefore so does its imaginary part ).

I'm not sure how to do the second part of the question. If this series were the Fourier series of asquare-integrable function then you could use the Parseval theorem to say that converges (which it doesn't). But if you are only allowed to assume that the function is integrable then this idea won't work.