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\title{Sequences and series}
\date{\today}

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

\maketitle

\vspace{-.15in}

\pagestyle{empty}
\begin{flushleft}

Definitions:
\begin{itemize}

\item A \textbf{sequence} $\{a_n\}$ is a list of numbers: $a_1, a_2, a_3, a_4, \dots$

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\item A \textbf{series} is a sum of numbers: $\sum_{n=1}^{\infty} a_n  = a_1 + a_2 + a_3 + a_4 + \dots$.

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\item There are two sequences associated with the series $\sum_{n=1}^{\infty} a_n$:
\subitem The sequence of terms: $a_1, a_2, a_3, \dots$.
\subitem The sequence of \textbf{partial sums}: $s_1, s_2, s_3 \dots$ where $s_k = \sum_{n=1}^k a_n = a_1 + a_2 + a_3 + \dots+ a_k$.

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\item The \textbf{sum of a series} is the limit of the sequence of partial sums: $\sum_{n=1}^{\infty} a_n = \lim_{k \to \infty} s_k$.

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\item The series $\sum a_n$ is \textbf{absolutely convergent} if both $\sum a_n$  and $\sum \lvert a_n \rvert$ are convergent.


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\item The series $\sum a_n$ is \textbf{conditionally convergent} if $\sum a_n$ converges, but $\sum \lvert a_n \rvert$ diverges.

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\end{itemize}



Tests:
\begin{itemize}

\item \textbf{Test for divergence}: if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.

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\item \textbf{Integral test}: if $f$ is a positive, decreasing, continuous function on the interval $[N, \infty)$ and $a_n = f(n)$ for $n \geq N$, then the series $\sum a_n$  and the integral $\int_N^{\infty} f(x)\ dx$ both converge or both diverge.
% \subitem The error of $s_n$ as a approximation for $\sum_{k=1}^{\infty} a_k$: \[ s_n + \int_{n+1}^{\infty} f(x)\ dx \leq  \sum_{k=1}^{\infty} a_k \leq s_n + \int_{n}^{\infty} f(x)\ dx \]

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\item \textbf{Direct comparison}: suppose that $0 \leq a_n \leq b_n$ for all $n \geq N$.
\subitem If $\sum b_n$ converges, then $\sum a_n$ converges.
\subitem If $\sum a_n$ diverges, then $\sum b_n$ diverges.

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\item \textbf{Limit comparison test}: suppose that $a_n  > 0$ and $b_n >0$ for all $n \geq N$ and $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = L$.
\subitem If $0 < L < \infty$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge.
\subitem If $L=0$ and $\sum b_n$ converges, then $\sum a_n$ converges.
\subitem If $L = \infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

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\item \textbf{Ratio test}: let $\displaystyle L = \lim_{n \to \infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert$
\subitem If $L<1$, then the series $\sum a_n$ is absolutely convergent.
\subitem If $L>1$, then the series $\sum a_n$ is divergent.
\subitem If $L = 1$, then no conclusion can be drawn.

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\item \textbf{Root test}: let $\displaystyle L = \lim_{n \to \infty} \sqrt[n]{\lvert a_n \rvert}$
\subitem If $L<1$, then the series $\sum a_n$ is absolutely convergent.
\subitem If $L>1$, then the series $\sum a_n$ is divergent.
\subitem If $L = 1$, then no conclusion can be drawn.

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\item \textbf{Alternating series test}: the alternating series $\sum (-1)^n b_n$ converges if:
\begin{enumerate}[i)]
\item $0 \leq b_{n+1} \leq b_n$ for all $n \geq N$ and
\item $\lim_{n \to \infty} b_n = 0$,
\end{enumerate}

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\end{itemize}

% \newpage

Special series: 
\begin{itemize}
\item \textbf{Geometric series:} 
$\displaystyle \sum_{n=0}^{\infty} r^n = 1 + r + r^2 + r^3 + \dots = \frac{1}{1-r}$ if $\lvert r \rvert < 1$ (and diverges if $\lvert r \rvert \geq 1$).

\vspace{.1in}

\item The {\bf $p$-series} $\displaystyle \sum \frac{1}{k^p} = 1 + \frac{1}{2^p} + \frac{1}{3^p} + \frac{1}{4^p} + \dots$ is convergent if and only if $p >1$.

\vspace{.1in}

\item The {\bf harmonic series} $\displaystyle \sum \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} +  \frac{1}{4} + \dots$ is divergent ($p$-series with $p=1$).
\end{itemize}

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