\documentclass{amsart} 
\linespread{1.05}
\usepackage{amssymb}
\usepackage[brazil]{babel}
\usepackage[latin1]{inputenc}
\setlength{\textwidth}{17.5cm}
\setlength{\textheight}{56pc}
\setlength{\baselineskip}{1pc}
\setlength{\parindent}{2mm}
\setlength{\parskip}{0.4pc}
\setlength{\oddsidemargin}{-0.52cm}
\setlength{\evensidemargin}{-0.52cm}
\setlength{\topmargin}{-2pc}
\setlength{\topskip}{3pc}
\setlength{\unitlength}{.3cm}
\newcommand{\ih}{\'{\i}}
\newcommand{\ao}{\~ao}
\newcommand{\cao}{\c{c}\~ao}
\newcommand{\coes}{\c{c}\~oes}
\newcommand{\oes}{\~oes}
\newcommand{\N}{{\mathbb N}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Q}{{\mathbb Q}}
\begin{document}
\begin{center}

{\sc MAT0315  - Introdu\cao\ \`a An\'alise Real
\\  IME-USP, 2$^\circ$ Semestre de 2013\\ Quarta Lista}
\end{center}
\vskip.6cm

\noindent ({\bf 1}) (a) Dados arbitr\'arios $a_n\in\{0,1,2,3,4,5,6,7,8,9\}$, $n\in\N$, mostre que 
${\displaystyle \sum_{n=1}\frac{a_n}{10^n}}$ converge. O que isto tem a ver com expans\ao\ decimal?

\vskip.5cm

\noindent ({\bf 2}) Dados arbitr\'arios $a_n>0$, $n\in\N$, mostre que:

(a) se ${\displaystyle \sum_{n=1}^{\infty}a_n}$ converge, ent\ao\ 
${\displaystyle \sum_{n=1}^{\infty}\frac{a_n}{1+a_n}}$ converge;

(b) ${\displaystyle \lim_{n\to\infty} a_n=0}$ se, e somente se, 
${\displaystyle \lim_{n\to\infty}\frac{a_n}{a_n+1}=0}$;

(c) se ${\displaystyle \lim_{n\to\infty} a_n=0}$ e ${\displaystyle \sum_{n=1}^{\infty}a_n}$ diverge, ent\ao\  ${\displaystyle \sum_{n=1}^{\infty}\frac{a_n}{1+a_n}}$ diverge;

(d) ${\displaystyle \sum_{n=1}^{\infty}a_n}$ converge se, e somente se, 
${\displaystyle \sum_{n=1}^{\infty}\frac{a_n}{1+a_n}}$ converge.

\noindent {\small Dicas: (a) compara\cao; (b) inverta a fun\cao\ $f(t)=t/(t+t)$; (c) compara\cao, usando que $a_n<1$ para todo $n$ suficientemente grande; (d) o caso n\ao\ coberto por (a) ou (c) segue do crit\'erio do termo geral, junto com o (b).}


\vskip.5cm

\noindent ({\bf 3}) Mostre que
\[
(\mbox{a})\ \sum_{n=2}^{\infty}\frac{n-1}{n!}=1,\ \ \ \ 
(\mbox{b})\ \sum_{n=2}^{\infty}\frac{n^2-2}{n!}=3,\ \ \ \ 
(\mbox{c})\ \sum_{n=1}^{\infty}\frac{n}{(n-1)!}=2e.
\]
{\small Dicas: No (a) e no (b), calcule algumas somas parciais, adivinhe a f\'ormula geral e demonstre-a por indu\cao. No (c), use o (b) e o valor da soma de uma s\'erie not\'avel, calculado em sala.} 

\vskip.5cm

\noindent ({\bf 4}) Determine para que valores de $x$ convergem absolutamente, convergem condicionalmente, ou divergem as s\'eries:
\begin{center}
(a) ${\displaystyle \sum_{n=1}^{\infty}\frac{4^n}{n^2}x^n}$,\ \ \ \ \ \ \ \ \ 
(b) ${\displaystyle \sum_{n=1}^{\infty}\frac{4^n}{n}x^n}$,\ \ \ \ \ \ \ \ \  
(c) ${\displaystyle \sum_{n=1}^{\infty}\frac{n}{4^n}x^n}$,
\end{center}

\begin{center}
(d) ${\displaystyle \sum_{n=1}^{\infty}n!\,x^n}$,\,\ \ \ \ \ \ \ \ \ 
(e) ${\displaystyle \sum_{n=1}^{\infty}\frac{n!}{n^n}x^n}$,\,\ \ \ \ \ \ \ \ \  
(f) ${\displaystyle \sum_{n=1}^{\infty}4^nx^{n^{2}}}$.
\end{center}


\end{document} 