\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Under the hypotheses of the integral
test, if $a_n=f(n)$ then for any
positive integer $N$, 
$\sum\limits_{N+1}^\infty a_n\le \int\limits_N^\infty f(x)\,dx$.

\medskip 

\noindent a) How large does $N$ have to be to ensure that

(1) $\sum\limits_{n=1}^N {1\over{n^5}}$ is within $10^{-6}$ of
$\sum\limits_{n=1}^\infty {1\over{n^5}}$? 

(2) $\sum\limits_{n=1}^N ne^{-n^2}$ is within $10^{-6}$ of
$\sum\limits_{n=1}^\infty ne^{-n^2}$? 

\medskip 

\noindent b) Get a decimal approximation for the sum of one of the
series with error less than $10^{-6}$. 

\vfil\eject\end