If we build the sum of two power series , $\sum a_kz^k$, $\sum b_kz^k$ both with radius of convergence 1, why does the resulting power series: $\sum (a_k+b_k)z^k$ also have RoC 1?
Is it because : $|\frac{a_k}{a_{k+1}}| = 1 , |\frac{b_k}{b_{k+1}}|=1$
$|\frac{a_k+b_k}{a_{k+1}+b_{k+1}}|$
that doesnt really do anything T_T .... If you see how to show this, please do tell.