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{\bf University of Notre Dame }\\
{\bf Math. 166, Final Examination (May 8,1997) }
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{\bf Instructor:  Hong-Ming Yin,\hspace{1cm} Duration: 2 hours.}

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[30] 1. The following statements may be true or false. If it is true, give a short proof, if not, give a counterexample
or give a true statement.
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(a) Let $f(x)$ and $g(x)$ be defined on $[0, \infty)$ and
\[\lim_{x\rightarrow +\infty}f(x)=+\infty,\,\lim_{x\rightarrow +\infty}g(x)=+\infty;\]
Then
\[ \lim_{x\rightarrow +\infty}[f(x)-g(x)]=0.\]

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(b) Let $z$ be a complex number and $i$ be the imaginery unit. Then 
\[  e^{z+2n\pi i}= e^{z} \hspace{1cm} \mbox{for all integer $n$.}\]

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(c) Let $\{a_n\}$ be a non-negative and monotone decreasing sequence. Moreover, \[ \lim_{n\rightarrow \infty}a_n=0\]
then  the series $\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{n}}$ converges.

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(d) For all $x\in (-1,\infty)$,
\[ log(1+x) =\sum_{n=1}^{\infty} (-1)^{n-1}x^n.\]

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(e) The function sequence $\{f_n(x)\} =\{ log (1+x^n) \} $ converges uniformly on $[0,1]$.

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(f) The following identity holds for all $x$:
\[ \mbox{arcsin} x+\mbox{arccos} x= 1.\]

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[20] 2.(a) Find the values of $a$ and $b$ such that
\[ \int_{2}^{+\infty} \left[ \frac{2x^{2}+bx+a}{x(2x+a)} -1\right] dx=1.\]
(b)  Prove the improper integral
\[ \int_{1}^{\infty}\frac{  x^s }{e^{x}}dx\]
is convergent for any fixed $s\in (-\infty, +\infty)$.



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[20] 3. For each of the following power series, determine the set  of all real $x$ for which
the series  converges.
\begin{eqnarray*}
(a) & & \sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{log(1+n)}} x^n;\\
(b) & & \sum_{n=1}^{\infty}\frac{n!}{n^{n}} x^n
\end{eqnarray*}



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[20] 4.  Find the Taylor's series centered at $0$ of the following functions and
indicate the radius of convergence for the series:\\
(a) $f(x)= sin{2x} $\\
(b) $g(x) = \frac{1}{2-x-x^{2}}$ 


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[10] 5. Let $f(x)=e^{-\frac{1}{x^{2}}}$ if $x\neq 0$ and $f(0)=0$.\\
(a) Prove that for every integer $m>0$, 
\[ \lim_{x\rightarrow 0} \frac{f(x)}{x^{m}}=0;\]
(b) Prove $f^{(n)}(0) =0$ for all $n\geq 1$.










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