Let . We have for :
Now show that for all integer we have .
I have the following problem: Prove that the power series \sum \frac {z^n}{n} converges at every point of the unit circle except z=1.
The part with z=1 I already did it, but I have problems with the rest. Is there a way to prove the convergence using Fourier coefficients?
Thanks.
The Abel test for convergence of a series extablishes that...
a) given a sequence such that the partial sum is bounded...
b) given a sequence so that is decreasing and ...
... then the series converges. In Your case satisfies to criterion b), so that we have to verify thwe criterion a) for the sequence . The partial sum is...
(1)
... so that is...
(2)
... and for it is evident that (2) is bounded. Then for Abel's test the series converges if ...
Kind regards