Mathematics

Problem

Let

h(x)=\displaystyle \sum_{n=0}^{\infty}\frac{n!}{(2n+2)!}(x-3)^{2n+1}

. (a) Determine the radius of convergence of the series for

h

. (b) Obtain a power series centered at

x=3

for an antiderivative

H(x)

h

with

H(3)=0

. (c) Using the series for

H

, evaluate

\displaystyle \int_{3}^{3+\tfrac{1}{4}} h(x)\,dx

as an infinite series and write the first three nonzero terms. (d) Provide an upper bound for the magnitude of the truncation error that results from using the first three nonzero terms in part (c). Present your reasoning clearly. a. Part (a) b. Part (b) c. Part (c) d. Part (d)

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