A power series is defined by for real values of .

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

Problem

A power series is defined by

P(x)=\displaystyle \sum_{n=0}^{\infty} \frac{n^2+1}{4^n}x^{n}

for real values of

x

. (1) Determine the interval of

x

for which

P(x)

converges absolutely. (2) Determine the values of

x

(if any) for which

P(x)

converges conditionally but not absolutely, and provide justification. (3) Combine your results from parts (a) and (b) to state the full set of real numbers for which

P(x)

converges.

Related theory

Infinite Series
Power Series

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