Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 1 / 4D / Solution

Use these standard facts about a power series: it is absolutely convergent at every point strictly inside its radius of convergence and divergent at every point strictly outside it. Also, the triangle inequality implies that a sum or difference of two absolutely convergent series is absolutely convergent.

Let $T$ be the radius of convergence of the coefficientwise sum. For $|z|<\min (R,S)$,Hence $T\ge \min (R,S)$.

Suppose $R<S$. If $T>R$, choose a finite $\rho$ with $R<\rho <\min (T,S)$ and choose $z$ of complex modulus $\rho$. The sum series and the $b_n$ series both converge absolutely there, socontradicting $\rho >R$. Thus $T=R$. Interchanging the series covers $S<R$, proving the sum of power series with unequal radii of convergence rule:The argument also covers a zero radius or an infinite larger radius. When the two radii are equal, cancellation can enlarge the radius, so the inequality of the radii matters.