Mathematics

Problem

For every real number

x

satisfying

|x|<3

, define the series

S(x)=\displaystyle\sum_{n=0}^{\infty} (-1)^{n}\left(\dfrac{x}{3}\right)^{n}

. Let

I=\displaystyle\int_{0}^{1} S(x)\,dx

. (a) Rewrite

S(x)

as a single elementary function of

x

valid for

|x|<3

. Show all steps that justify your closed‐form expression. (b) Evaluate the integral

I

exactly. (c) Explain why term-by-term integration of the original series

\sum_{n=0}^{\infty} (-1)^{n}\left(\dfrac{x}{3}\right)^{n}

on the interval

[0,1]

is permissible.