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\vspace{-1in}
\begin{center}
 \large\bf \m Math 365,  Test \# 3
 $\left.\hspace{2em}\oval(80, 40)\right/ 100$
 \end{center}
\begin{flushright}
Dec 6,  1996\\
Instructor: Bei Hu
\end{flushright}

\hspace{.5\textwidth} Name:\hrulefill


1. (25 points). True or False. (No proof or counter example needed).

(a). A continuous function defined on
a bounded and closed set of $R^k$ is always uniformly continuous.
\vspace{1in}

(b). Suppose $a_n>0$.  If $\dis 
\limsup_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} >1 $,
 then the series $\dis\sum a_n$ is divergent.
 
\vspace{1in}

(c). In metric spaces, if the inverse function of a continuous 
function exists, then it must be continuous.
\vspace{1in}

(d). A convergent series is always absolutely convergent.
\vspace{1in}

(e). If $f:X\longrightarrow Y$ is a continuous function and 
$E\subset Y$ is a connected set, then $f^{-1}(E)$ is connected.

\newpage
2. (20 points).  

(a). State PRECISELY the definition for the radius of 
convergence for a power series.
\vspace{1.8in}

(b). State PRECISELY the formula of Summation by parts.
\vspace{1.8in}

(c). State PRECISELY the intermediate value theorem.
\vspace{1.8in}

(d). State PRECISELY two equivalent definitions for a 
continuous function.
\vspace{1.8in}
\newpage


3. (10 points). Determine the radius of convergence for 
the following power series.

(a). $\dis \sum_{n=0}^\infty (2+n^3) z^n$.
\vspace{3.6in}

(b). $\dis \sum_{n=0}^\infty  \frac{4^n}{n^7} z^n$.

\newpage


4. (10 points). Determine the convergence for 
the following series, indicating the theorem that you used.

(a). $\dis \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$.
\vspace{3.6in}

(b). $\dis \sum_{n=1}^\infty  \frac{2^n n!}{n^n}$.

\newpage



5. (8 points)
  Let $f$ be a real function on a metric space $X$.
Prove that the zero set of $f$, defined by
 $Z(f)=\{p\in X\bigg| f(p) = 0\}$,
is a closed set in $X$.

\vspace{4in}



6. (7 points)
Suppose that $a_n\ge 0$ and $\dis\sum a_n$ converges.
Prove that $\dis \frac{\sqrt{a_n}}{n}$ converges.


\newpage

7. (10 points)
Let $\dis f(x) = \sin \bigg(\frac1x\bigg)$ and $g(x) = xf(x)$.

(a). Is $f(x)$ continuous on $(0,1)$?

(b). Is $g(x)$ continuous on $(0,1)$?

(c). Is $f(x)$ uniformly continuous on $(0,1)$?

(d). Is $g(x)$ uniformly continuous on $(0,1)$?

Justify your answers.


\newpage

8.  (10 points). Let $f$ be a continuous function from the 
metric space $X$ to the metric space $Y$. 

(a). Prove that for $E\subset X$, $f(\bar E) \subset \bar{f(E)}$.

(b). Is the equality $f(\bar E) = \bar{f(E)}$ valid? (Prove or give
a counter example).

(c). Is the equality in (b) valid if $X=(-\infty,\infty)$ and
$E$ is bounded? (Prove or give
a counter example).
\newpage

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