\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Under the hypotheses of the integral
test, if $a_n=f(n)$ and if
$s_n=a_1+a_2+\cdots+a_n=\sum\limits_{j=1}^na_j$, then $\int_1^n
f(x)\,dx\le s_n\le a_1+\int_1^n f(x)\,dx$ for each positive
integer $n$.  

\noindent For the harmonic series $\sum\limits_{j=1}^\infty{1\over
j}$, this implies $\ln n \le 1 + {1\over 2}+{1\over 3}+\cdots+{1\over
n}\le 1+\ln n$ for each positive integer $n$.

\noindent a) Find the analogous inequalities for the series
$\sum\limits_{j=1}^\infty {1\over{\sqrt j}}$ and for the series
$\sum\limits_{j=2}^\infty {1\over{j\ln j}}$. 

\medskip 

\noindent b) Estimate the sum of the first $10^{10}$ terms of the
series, in each of the three cases. Then estimate the sum of the first
$10^{100}$ terms.

\medskip 

\noindent c) Of the three series, which diverges the fastest? the slowest?  


\vfil\eject\end

