u have Cauchy - Hadamard Th. saying that :
let the be radius of convergence of the power series , then it is worth ::
then if , then radius is which means that power series are converging just in point (all power series are converging at least at point x=0)
or if then radius is which means that series converge for all .
P.S. you can depending on the situations use another formula for calculating radius of convergence (less work sometimes)
now if i understand your question ... you are referring to the Th and need the proof of it which says :
let the power series is converging for some value . Than series are absolutely convergent for all for which is and that series uniformly convergent for all for which where we have that .
let's assume that for series converge. That means that common member of the series must strive for zero, and thus he as a convergent sequence is limited. So there is constant such that for all this is true
now let the than is true that
because than , series as the sum of the geometrical series is converging, which now means that and series is converging, and that is absolutely converge.
now we show second part
let where Based on the proof above we know that series absolutely converge. As for the selected is valid ::
using Weierstrass criteria we have uniform convergence of the series