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\begincenter{}\endcenter







\vspace{-1in}
\begin{center}
 \large\bf \m Math 365,  Final Test 
 $\left.\hspace{2em}\oval(80, 40)\right/ 150$
 \end{center}
\begin{flushright}
Dec 18,  1996\\
Instructor: Bei Hu
\end{flushright}

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


1. (35 points). True or False. (No proof or counter example needed).

(a). A finite set in a metric space is compact.
\vspace{.8in}

(b). If $E\subset X$ is a connected set and $f$ is a continuous function
on $X$, then $f(E)$ is also a connected set.
 
\vspace{.8in}

(c). If a function $f(x)$ is differentiable at a point $x=a$,
then $f(x)$ is also continuous at the point $x=a$.
\vspace{.8in}

(d). A nonempty perfect set in a metric space is uncountable.
\vspace{.8in}

(e). If a sequence converges, then any rearrangement of the
series also converges.
\vspace{.8in}

(f). In a metric space $X$, let $\{F_n\}$ be  a sequence of
 nonempty compact sets such that $F_n \supset F_{n+1}$. Then
 the intersection $\dis\bigcap_{n=1}^\infty F_n$  is nonempty.
\vspace{.8in}

(g). In a metric space, a continuous function maps a compact set
 to a compact set.


\newpage
2. (25 points).  

(a). State PRECISELY the Lagrange mean value theorem.
\vspace{1.5in}

(b). State PRECISELY the Taylor's formula (Assumptions and conclusions).
\vspace{1.5in}

(c). State PRECISELY Weierstrass theorem.
\vspace{1.5in}

(d). State PRECISELY the root test for a series.
\vspace{1.5in}

(e). State PRECISELY the least upper bound principle.
\vspace{1.5in}

\newpage


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

(a). $\dis \sum_{n=0}^\infty \frac1{(2n)!} z^n$.
\vspace{3.6in}

(b). $\dis \sum_{n=0}^\infty
\frac{3^n}{(1+4n^3+6n^5+100n^{99})} 
z^n$.

\newpage


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

(a). $\dis \sum_{n=1}^\infty\frac{1}{\sqrt{n}}
 \sin\Big(\frac{\pi}4 n\Big)$.
\vspace{3.6in}

(b). $\dis \sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}n $.

\newpage



5. (10 points)
 Suppose that $\dis C_0+\frac{C_1}2+\cdots+\frac{C_{n-1}}{n}
  + \frac{C_n}{n+1} = 0$,
where $C_0,\cdots, C_n$ are real constants. Prove that the
 equation
$$ C_0 + C_1 x +\cdots+ C_{n-1} x^{n-1} + C_n x^n = 0 $$
has at least one real root between $0$ and $1$.

\newpage



6. (10 points) $f$ is a real valued function defined on $R^1$.
Suppose that there exists a constant $A<1$ such that
$|f'(t)|\le A$ for all real $t$.

Define $x_0 = 1$ and
$$ x_{n+1} = f(x_n) $$
for $n=1,2,3,\cdots$. Prove that

(a) the limit $\dis x=\lim_{n\rightarrow\infty} x_n$ exists, 

(b)
the $x$ obtained in (a) satisfies $f(x) = x$.

\newpage

7. (10 points) Find the limits. Quote all theorems that you used.

(a). $\dis \lim_{x\rightarrow 0} \frac{\cos(x^{49}) - 1}{ x^{98}} $

\vspace*{3.5in}

(b). $\dis \lim_{n\rightarrow \infty}{\bigg.}^n\hspace{-.5ex}
 \sqrt{\frac{n!}{n^n}}$

(Hint: is there a theorem  about the relationship between root and ratio?)

\newpage

8.  (10 points). Suppose that $\dis\lim_{n\rightarrow\infty}
 a_n = a$ and let 
$$ \sigma_n = \frac{a_1+a_2+\cdots+a_n}{n}.$$
Prove that $\dis\lim_{n\rightarrow\infty} \sigma_n = a$.
\newpage

9.  (10 points). Suppose that the function $f: X\rightarrow R^1$
is continuous. The level set at level $\beta$ ($\beta$ is a real
number ) is
defined as $L_\beta=\bigg\{p\in X\bigg| f(p) = \beta \bigg\}$.

Prove that for any real $\beta$, the level set $L_\beta$ is
a closed set.

\newpage

10.  (10 points).

(a)  Please explain: We know that the set $[0,\infty)$ is not compact.
However, here is a ``proof''.

{\bf Proof:} Define the following metric $d(x,y) = \dis
\frac{|x-y|}{1+|x-y|}$. (This is a metric, and you don't have to
verify this.)  With this metric,
 the set $[0,\infty)$ is a closed and bounded
set in the Euclidean space $R^1$. Therefore, by Heine-Borel 
theorem, $[0,\infty)$ is a compact set. 

What's wrong with this proof?


\vspace*{3in}

(b) Suppose that $f: [a,b]\rightarrow R^2$ is a
vector valued function which is differentiable
everywhere on $[a,b]$. 
Is the mean value theorem valid for $f$? namely, there exists
$\xi \in (a,b)$ such that $ f(a) - f(b) = (b-a) f'(\xi)$.

Prove if it is true. Give a counter example if it is false.


\newpage

11.  (10 points). Is the following statement true? Prove
if it is true, give a counter example if it is false.

Suppose that $\{x_n\}$ is a Cauchy sequence on the metric
space $X$ and $f$ is a continuous function from $X$ to $Y$. Then the
sequence $\{f(x_n)\}$ is a Cauchy sequence on $Y$.
\newpage


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