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\title{Series I}
\date{October 15, 2020}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{
Define a sequence $\{ s_n \}_{n=1}^{\infty}$ as follows:
\[ s_1 = \frac{1}{2} \text{ and } s_{n+1} = s_n + \frac{1}{2^{n+1}} \]
\begin{enumerate}[a)]
\item Calculate $s_2$, $s_3$, $s_4$, and $s_5$.

\vspace{.1in}

\item Find a (non-recursive) formula for $s_n$.

\vspace{.1in}

\item Calculate $\lim_{n \to \infty}s_n$.

\vspace{.25in}

\end{enumerate}
}

\prob{
Define a sequence $\{ s_k \}_{k=1}^{\infty}$ as follows:
\[ s_k = \sum_{n=1}^{k} \frac{2}{n(n+2)} \]
\begin{enumerate}[a)]
\item Calculate $s_1$, $s_2$, and $s_3$. (This will be less helpful than it was in the last problem).

\vspace{.1in}

\item Find a partial fraction decomposition for $\displaystyle \frac{2}{n(n+2)}$.

\vspace{.1in}

\item Use your partial fraction decomposition to find a formula for $s_k$.

\vspace{.1in}

\item Calculate the limit $\lim_{k \to \infty}s_k$.

\vspace{.25in}

\end{enumerate}
}

\prob{
Decide if each of the following sequences converges or diverges. It's okay to guess, by try to explain what you're thinking.
\begin{enumerate}[a)]
\item $\displaystyle s_1= \frac{1}{3}$ and $\displaystyle s_{n+1} = s_n + \frac{1}{3^{n+1}}$

\vspace{.1in}

\item $\displaystyle s_1= 1$ and $\displaystyle s_{n+1} = s_n + (-1)^{n+1}$

\vspace{.1in}

\item $\displaystyle s_1=1$ and $\displaystyle s_{n+1} = s_n + \frac{n+1}{n}$

\vspace{.1in}

\item $\displaystyle s_1= 1$ and $\displaystyle s_{n+1} = s_n + \frac{1}{n+1}$
\end{enumerate}
}

\end{document}