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\lhead{\bf Advanced Calculus (MAA 4212)}
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\rhead{Due Wednesday 4/3}
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%\bigskip\noindent\emph{Reminder: Midterm 3 is on Wednesday 4/23.}

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\begin{enumerate}

\Problem Find a power series representation for the \emph{error function},
\[
	\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2)\, dt.
\]

\Problem Suppose that $(a_k)$ and $(b_k)$ are sequences of real numbers and that both of the series
\[
	\sum_{k\ge 0} a_k^2 \ \mbox{ and } \sum_{k\ge 0} b_k^2
\] 
converge. Prove that the series
\[
	\sum_{k\ge 0} a_k b_k
\]
also converges.
\emph{Suggestion:} use the inequality $2|xy|\le x^2+y^2$, or apply the Cauchy--Schwarz inequality to the partial sums.


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