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\lhead{\bf Advanced Calculus (MAA 4212)}
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\rhead{Due Friday 1/25}
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\begin{enumerate}

\Problem Let $f_n\st (-1,1)\to\mathbb{R}$ be given by $x^n$ for all $n\ge 0$. Prove that
\begin{enumerate}
\item[(a)] the sequence $(f_n)$ converges pointwise (and determine the function it converges to) and
\item[(b)] the sequence $(f_n)$ does not converge uniformly.
\end{enumerate}

\Problem Suppose that the function $f\st [0,1]\to\mathbb{R}$ is continuous and define the functions $g_n\st [0,1]\to\mathbb{R}$ by $g_n(x)=x^nf(x)$. Prove that
\begin{enumerate}
\item[(a)] if the sequence $(g_n)$ converges uniformly then $f(1)=0$ and
\item[(b)] if $f(1)=0$ then the sequence $(g_n)$ converges uniformly.
\end{enumerate}

\end{enumerate}

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