When is represented by the power series with radius of convergence

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

When f(x)f(x) is represented by the power series ∑n=0∞cn (x−3)n\sum_{n=0}^{\infty} c_n\,(x-3)^n with radius of convergence RR, a student integrates the series term-by-term to obtain g(x)=∑n=0∞cnn+1(x−3)n+1+Cg(x)=\sum_{n=0}^{\infty} \dfrac{c_n}{n+1}(x-3)^{n+1}+C. Which statement about the convergence of the series for g(x)g(x) is correct? (a) The radius of convergence of the series for gg is R+1R+1. (b) The series for gg converges for every real xx (that is, its radius of convergence is infinite). (c) The radius of convergence of the series for gg equals RR. (d) The radius of convergence of the series for gg is R−1R-1.

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