\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} The following problem is quoted
directly from D.~Knuth, {\sl The Art of Computer Programming}, the
standard (multivolume) reference for theoretical computer
science. This problem is in volume 1, {\sl Fundamental Algorithms}.

\medskip

If $f(x)= \sum_{k \ge 0} a_k x^k$, and this series converges for
$x=x_0$, then show that
$$\sum_{k\ge 0} a_k x_0^k H_k = \int_0^1 
{{ f(x_0)-f(x_0 y)}\over {1-y}} dy.$$
Here the numbers $H_k$ are defined to be the partial sums of the
harmonic series: $H_0=0$; 
$H_k = 1+ {1\over 2} + {1 \over 3} + \ldots + {1 \over k}$ for $k\ge1$.

\smallskip

\noindent {\bf Hint} Expand everything on the right side in a power
series in $x_0$.  What is the coefficient of $x_0^k$? The formula for
the sum of a {\it finite} geometric series will be useful.

\vfil\eject\end