Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 5 / a / iii / Solution

Take $0\le r<R$. At the point $r$, the power series has absolute convergence, so ${\sum }_{n=0}^{\infty }|a_n|r^n<\infty$. On the whole interval $[-r,r]$,The Weierstrass M-test with $M_n=|a_n|r^n$ proves uniform convergence on $[-r,r]$. It also gives the explicit tail estimateFor $r=0$ the interval consists only of zero, and all positive-degree terms vanish. If $R=\infty$, the same argument applies to every finite $r$; if $R=0$, there is no $r\ge 0$ with $r<R$.