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\begin{document}
\begin{center}

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

Este trabalho \'e uma aplica\cao\ da f\'ormula
\begin{equation}\label{formula}
\lim_{n\to\infty} \left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right)=\gamma,
\end{equation}
onde $\gamma$ \'e uma constante maior do que $1/2$ e menor do que $1$, para provar a f\'ormula 
\[
1+\frac{1}{2}-\frac{2}{3}+
\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+
\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+
\frac{1}{10}+\frac{1}{11}-\frac{2}{12}+
\cdots=\ln 3.
\]


\[
\cdots\cdots\cdots
\]

Considere $a_n$, $n\in\N$, definidos por
\[
a_n=\left\{\begin{array}{l}
\frac{1}{n}\ \mbox{se}\ n\ \mbox{n\ao\ \'e divis\'ivel por}\ 3\\
-\frac{2}{n}\ \mbox{se}\ n\ \mbox{\'e divis\'ivel por}\ 3\\
 \end{array}\right. 
\]
e denote por $s_n$ a soma parcial $s_n=a_1+a_2+\cdots+a_n$.

\vskip.3cm
(a) Use (\ref{formula}) para mostrar que existe uma sequ\^encia $(r_n)$ tal que $r_n\to 0$ e 
\[
1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\ln n+\gamma+r_n,\ \forall n\in \N.
\]

\vskip.3cm
(b) Mostre por indu\cao\ que 
\[
s_{3n}=\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{3n}\right)-
\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right),\ \forall n\in\N.
\]

\vskip.3cm
(c) Use os dois itens anteriores para mostrar que 
${\displaystyle \lim_{n\to\infty} s_{3n}=\ln 3}$.

\vskip.3cm
(d) Usando que $s_{3n+1}=s_{3n}+a_{3n+1}$ e que $s_{3n+2}=s_{3n}+a_{3n+1}+a_{3n+2}$, mostre que ${\displaystyle \lim_{n\to\infty} s_{3n+1} =\lim_{n\to\infty} s_{3n+2}=\ln 3}$ e assim conclua que 
\[\sum_{n=1}^{\infty}a_n=\ln 3.\]



\end{document}
