\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Verify that the infinite series
$\sum\limits_{n=2}^\infty {{(-1)^n}\over{n\left(\ln(n)\right)^2}}$
converges. A computer gives the approximate value .84776 to 5 digit
accuracy for the sum of this series. Find a specific partial sum which
is guaranteed to give this number to 5 digit accuracy. Give evidence
supporting your assertion.

\medskip

\noindent b) Verify that the infinite series $\sum\limits_{n=2}^\infty
{{1}\over{n\sqrt{\ln(n)}}}$ diverges.  A computer gives the
approximate value of 4.74561 for the $10,\!000^{\rm th}$ partial
sum. Are the partial sums of this series unbounded? If yes, find a
specific partial sum which is guaranteed to be greater than 100. Give
evidence supporting your assertion.

\medskip

\noindent {\bf Comment} In neither case is the ``best possible''
partial sum requested. Supporting evidence must be presented for the
two partial sums given.

\vfil\eject\end