The Maclaurin series for converges for all real numbers

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

Problem

The Maclaurin series for sin⁡x\sin x converges for all real numbers. Which statement best explains why the power series for sin⁡(2x3)\sin(2x^{3}) also converges for every real xx? (a) Because the Maclaurin series for sin⁡x\sin x converges when ∣x∣<1|x|<1, substituting 2x32x^{3} gives ∣2x3∣<1|2x^{3}|<1, so the series converges only for ∣x∣<1/23|x|<\sqrt[3]{1/2}. (It therefore fails to converge for larger ∣x∣|x|.) (b) The series for sin⁡(2x3)\sin(2x^{3}) is actually a finite polynomial, so convergence is automatic. (c) Substituting 2x32x^{3} into a series that converges for all real xx produces another series whose terms remain finite for every real xx, so its interval of convergence is unchanged. (d) Since (2x3)2n+1(2x^{3})^{2n+1} grows faster than (2n+1)!(2n+1)!, the series diverges for large ∣x∣|x|; therefore it converges only for ∣x∣<1|x|<1.

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