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\begin{document}

\centerline{\textbf{Sample Midterm Questions}}
\centerline{\textbf{Math 127B}. \textbf{Winter, 2005}}
\centerline{\emph{Closed Book. No calculators.
Give complete proofs of all your answers.}}
%\centerline{\emph{You can use any standard theorem
%provided you state it carefully.}}

\bigskip\bigskip
\noindent
\textbf{1.} Consider the sequence $(f_n)$ of functions $f_n : \Rl\to \Rl$
defined by
\[
f_n(x) = \frac{nx}{\sqrt{1 + n^2 x^2}}.
\]
Find the pointwise limit of this sequence as $n\to \infty$. Does the sequence
converge uniformly on $\Rl$? Justify your answer.



\bigskip
\noindent
\textbf{2.}  Let
\[
f_n(x) = \frac{nx + \sin(nx^2)}{n}.
\]
Prove that the following limit exists, and compute its value:
\[
\lim_{n\to\infty} \int_0^1 f_n(x) \, dx.
\]



\bigskip\bigskip
\noindent
\textbf{3.} Prove that the following series
\[
f(x) = \sum_{n=1}^\infty \frac{n^2 + x^4}{n^4 + x^2}
\]
converges to a continuous function $f: \Rl\to \Rl$.  



\bigskip
\noindent
\textbf{4.} Determine the radius of convergence $R$ of the power series
\beann
f(x) &=& \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1}\\
&=& x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots.
\eeann
Where does the series converge? Prove that
\[
f^\prime(x) = \frac{1}{1+x^2}\qquad\mbox{in $|x| < R$}.
\] 



\bigskip
\noindent
\textbf{5.} Suppose that $(f_n)$ is a sequence of functions $f_n : [-1,1] \to \Rl$
that converges uniformly on $[-1,1]$ to a function $f: \Rl \to \Rl$. If the limit
\[
\lim_{x\to 0} f_n(x) = a_n
\]
exists for each $n\in \Nl$, and the limit
\[
\lim_{n\to\infty} a_n = a
\]
exists, prove that $\lim_{x\to 0} f(x)$ exists, and
\[
\lim_{x\to 0} f(x) = a
\]
Give a counter-example to show that this result need not be true if $(f_n)$
converges to $f$ pointwise, but not uniformly.
\end{document}