\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Consider the following improper integrals:
$$(1)\ \int_1^{\infty} {{\ln x} \over {x}} \, dx\,;\quad
(2)\ \int_1^{\infty} {{\ln x} \over {\sqrt x}} \, dx\;\quad
(3)\ \int_1^{\infty} {{\ln x} \over {x^3}} \, dx\,.
$$

\noindent a) Graph the integrands. Determine which integrals are
larger than the others for large $x$.

\medskip 

\noindent b) Do integrals (1) and (2) converge or diverge?  

\medskip 

\noindent {\bf Hint} Try a substitution to evaluate one of the
integrals.

\medskip 

\noindent c) Explain why your knowledge of integrals (1) and (2) does
not help you decide whether integral (3) converges.

\medskip 

\noindent d) Explain why ${{{\ln x}\over{x^3}}<{{1}\over{x^2}}}$ for
all $x>1$. Use this to decide whether integral (3) converges.

\vfil\eject\end

