\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} A computer reports the following
information:

\smallskip

\centerline{$\sum\limits_{j=1}^{10}{1\over{j^3+2j^2+j}}\approx 0.35105\,;\quad
\sum\limits_{j=1}^{100}{1\over{j^3+2j^2+j}}\approx 0.35501\,;\quad
\sum\limits_{j=1}^{1000}{1\over{j^3+2j^2+j}}\approx 0.35506\,.$}

\noindent This suggests that
$\sum\limits_{j=1}^{\infty}{1\over{j^3+2j^2+j}}$ converges and that
its sum is 0.355 (to 3 decimal places). Explain the details in the
following outline of a verification of this statement. 

\medskip

\noindent a) The series has all positive terms. Therefore if the
infinite tail $\sum\limits_{j=101}^{\infty}{1\over{j^3+2j^2+j}}$
converges and has sum less than .001, the omitted terms after the
first 100 of the whole series won't matter to 3 decimal places.
 
\medskip

\noindent b) Overestimate the infinite tail
$\sum\limits_{j=101}^{\infty}{1\over{j^3+2j^2+j}}$ by the infinite
tail of a simpler series. Then compare the infinite tail of this
simpler series to a simple improper integral. Use a diagram to help
explain the comparison. Compute the improper integral.

\vfil\eject\end