Mathematics

Problem

For all real numbers

x

, the Maclaurin series for

e^x

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.

Define

S(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} \quad\text{and}\quad R(x)=1+x+\left(\frac{x^2}{2!}+\frac{x^3}{3!}\right)+\left(\frac{x^4}{4!}+\frac{x^5}{5!}\right)+\left(\frac{x^6}{6!}+\frac{x^7}{7!}\right)+\cdots,

where

R(x)

is formed by grouping the terms of

S(x)

into consecutive pairs after the first two. (a) Show that

S(x)

converges for all real

x

. (b) Evaluate

S(1)

. (c) Determine whether

R(1)

converges and, if it does, find its sum. (d) Explain the reasoning that leads from parts (a) and (b) to your conclusion in part (c).