## Proof of the Radius of Convergence Theorem

For a given power series, precisely one of the following may happen

1. the series converges only for 2. the series converges for all 3. there is some real number such that converges and converges absolutely if and diverges if The proof rests on the claim that if a power series is convergent for some on a circle with centre the origin, then it is absolutely convergent for all points inside the circle.

Note that if is convergent then so that for some Hence for and the series converges so the series converges by the comparison test.

Suppose that neither 1 or 2 above hold. Consider the set Since 1 does not hold there is some such that is convergent and hence the set is not empty. Moreover, is an interval (and not a set of discrete points) since and then Since 2 does not hold there is some such that is divergent and hence is divergent. Thus and so the interval of convergence has a finite, non zero, right hand end point (which may or may not be in ).

If then is convergent hence is absolutely convergent.

On the other hand, if then we can choose to satisfy then is divergent (since ) hence is divergent.