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Suppose that $\{a_{n}\}_{n \geq 0}$ is a sequence of positive real
numbers which decrease monotonically and which converge to zero.
Then by Theorem 3.43, the alternating series
\[
\sum_{n=0}^{\infty} (-1)^{n} a_{n}
\]
converges.

\begin{enumerate}
\item [(a)] With $\{a_{n}\}$ as above, suppose that $\displaystyle
\sum_{n=0}^{\infty} (-1)^{n} a_{n} = L$.  For any integer $k \geq 0$,
let $\displaystyle s_{k}=\sum_{n=0}^{k} (-1)^{n} a_{n}$ be the $k$th
partial sum.  Show that
\[
|L - s_{k}| \leq a_{k+1}.
\]
\item [(b)] The alternating series $\displaystyle \sum_{n=0}^{\infty}
\frac{(-1)^{n}}{n!}$ converges to $1/e$.  According to part (a), how
close is the partial sum $\displaystyle \sum_{n=0}^{6} (-1)^{n}/n!$ to $1/e$?

(You might want to compare with what a calculator gives you for the
actual difference $\displaystyle |1/e - \sum_{n=0}^{6} (-1)^{n}/n!|$.)
\end{enumerate}

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