Define for all in its interval of convergence.

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

Problem

Define

h(x)=\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\,(x-1)^{n}

for all

x

in its interval of convergence. (a) Determine the interval of convergence of

h(x)

, including a conclusion about each endpoint. (b) Find a power series centered at

x=1

for

h'(x)

and state its radius of convergence. (c) Evaluate

\displaystyle \lim_{x\to 1} \frac{h(x)}{x-1}

and interpret the result in terms of

h'(1)

. (d) Use the series for

h'(x)

to compute

h''(1)

. (e) Let

p(x)=h(x)+\dfrac{(x-1)^{2}}{2}

. Classify

x=1

as a local maximum, local minimum, or neither for

p

by analyzing appropriate derivatives obtained from your series work.