In rudin analysis text book, 3.44 theorem , it says about the convergence on the boundary of the circle of the power series, the theorem is roughly:-
Suppose the radius of convergence of $\Sigma c_nz^n $ and suppose the $c_n$ series is monotonically decreasing$c_n->0 $ as n-> infinity. Then $\Sigma c_nz^n $ converges at every point of mod z = 1 except possibly at z=1.
My question is :-
Why would not the same proof applies under the given hypothesis for when the radius of convergence is $<1$ ?
But if the proof is true for that case too then is not it a contradiction to 3.39 theorem (as there would be points mod z = 1 may not be z=1 such that the series converges that is the series is converging beyond the radius of convergence)(so where am I mistaking?)(saying about the existence of a radius of convergence) ?
Well and in 3.42 theorem ('partial sums of series of $a_n$ is bounded' and ' $ b_n$ decreases monotonically' and $b_n->0$ ' the $\Sigma a_nb_n$ converges.) saying about convergence of series of $a_nb_n$ do the series $a_n$ needs to converge ?