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\begin{document}

\centerline{\bf MAT 0315 - Introdu\cao\ \`a An\'alise Real - Turma 47} 

\centerline{\bf 2$^a$ Prova - 25 de novembro de 2013}

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\begin{center}
Cada quest\ao\ em branco vale 0,25.
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\noindent
{\bf Quest\ao\ 1:} Considere a sequ\^encia $(s_n)$ definida por 
$
{\displaystyle s_n=\sum_{k=1}^{n}k.k!}$, $n\in\N$.

\noindent
(a) Calcule $s_1$, $s_2$, $s_3$ e $s_4$.

\noindent
(b) Descubra, por inspe\cao, uma poss\'ivel f\'ormula geral para $s_n$ e demonstre, por indu\cao, sua validade.

\noindent
(c) Mostre que $\lim s_n=\infty$.

\newpage
\noindent
{\bf Quest\ao\ 2:} Decida se s\ao\ verdadeiras ou falsas as duas proposi\coes\ abaixo. Justifique sua resposta.

(a) Se, para todo $n\in\N$, $a_n>0$ e ${\displaystyle \frac{a_{n+1}}{a_n}\geq 1}$, ent\ao\ 
${\displaystyle \sum_{n=1}^{\infty}a_n}$ diverge. 

(b) Se, para todo $n\in\N$, $a_n>0$ e ${\displaystyle \frac{a_{n+1}}{a_n}\leq 1}$, ent\ao\ 
${\displaystyle \sum_{n=1}^{\infty}a_n}$ converge. 

\newpage
\noindent
{\bf Quest\ao\ 3:} Mostre que $\lim 2^{-n^2}n!=0.$

\newpage
\noindent
{\bf Quest\ao\ 4:} Mostre que a s\'erie ${\displaystyle \sum_{n=1}^{\infty}\frac{n^n}{n!}x^n}$

(a) converge absolutamente se $|x|<1/e$,

(b) converge se $x=-1/e$, 

(c) diverge se $|x|>1/e$.


\end{document}
