Consider an arbitrary power series S(x)= _ n=0 ^ a_ n x^ n with radius of convergence .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider an arbitrary power series S(x)=∑n=0∞anxnS(x)=\sum_{n=0}^{\infty} a_{n}x^{n} with radius of convergence R>0R>0. (a) Explain why S(x)S(x) converges absolutely for all xx satisfying ∣x∣<R|x|<R. (b) Provide a reasoning argument showing that the formal derivative series S′(x)=∑n=1∞n anxn−1S'(x)=\sum_{n=1}^{\infty} n\,a_{n}x^{n-1} also converges to S′(x)S'(x) for ∣x∣<R|x|<R. (c) Discuss how the radius of convergence of the differentiated series compares with that of the original series and explain what additional analysis is required to decide convergence at x=±Rx=\pm R. Give an example illustrating one possible outcome at an endpoint.

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