Let $\sum_{k=0}^{\infty}a_kx^k$ a power series with radius of convergence $R>0$

Question

• Show that if the Power Series diverges in $x=R$, then it doesn't uniformly converges in $[0,R]$
• Show that if the Power Series converges in $x=R$, then it does converges uniformly in $[0,R]$

This questions seems to me easy, just to put the concepts in practice, but i really don't got some notions about power series, then i would like to hear from you guys the fundamental concepts to resolve this questions.

Answer 1

Hints: 1. I assume you meant $[0,R)$ in the first problem. Suppose $\sum a_nx^n$ converges uniformly on $[0,R).$ Then the partial sums of $\sum a_nx^n$ are uniformly Cauchy on $[0,R).$ Show that this implies the partial sums of $\sum a_nR^n$ are Cauchy, hence $\sum a_nR^n$ converges, contradiction.

2. Suppose $\sum a_nR^n$ converges. Using summation by parts, show that $\sum a_nx^n$ is uniformly Cauchy on $[0,R].$ See Abel's theorem for inspiration.