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% A one-page summary of all the convergence
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% Copyright 2013 Aaron Decker
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\lhead{Aaron Decker, http://deckeraa.github.com}
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\section{Infinite Sequences and Series}
\subsection{Sequences}
A sequence is called \underline{monotonic} if it is either increasing or decreasing, not both. \\
A sequence is \underline{bounded above} if there exists $M$ such that $a_n \leq M$ for all $n\geq1$. \\
A sequence is \underline{bounded below} if there exists $m$ such that $a_n \geq m$ for all $n\geq1$. \\
Every bounded, monotonic sequence is convergent.

\subsection{Series}
\textbf{Geometric Series}: $\sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + \ldots$ \\
Convergent if $|r| < 1$, then sum is $\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}$ \\
Divergent if $|r| \geq 1$.

\textbf{Telescoping Series}: Write out several terms and look for a pattern of cancellation.

\textbf{Test for Divergence}: If $\lim_{n->\infty} a_n$ does not exist or $\lim_{n->\infty} a_n \neq 0$,
then $\sum^{\infty}_{n=1} a_n$ diverges. \\
WARNING: The Test for Divergence cannot prove that a series is convergent.

\subsection{The Integral Tests and Estimates of Sums}
\textbf{Integral Test}:\\
Preconditions: $f(x)$ is continuous, positive, and decreasing on $[1,\infty)$.\\
If $\int^{\infty}_{1} f(x) dx$ converges, then $\sum_{n=1}^{\infty} a_n$ converges.\\
If $\int^{\infty}_{1} f(x) dx$ diverges, then $\sum_{n=1}^{\infty} a_n$ diverges.

\textbf{P-Series Test}: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p>1$, diverges if $p \leq 1$ \\
Remainder Estimate for the Integral Test: $\int^{\infty}_{n+1} f(x) dx \leq R_n \leq \int^{\infty}_{n} f(x) dx$

\subsection{The Comparison Tests}
\textbf{Comparison Test}:\\
Preconditions: $ \sum a_n$ and $\sum b_n$ are series with positive terms. \\
If $\sum b_n$ is convergent and $a_n \leq b_n$ for all $n$, then $\sum a_n$ converges.\\
If $\sum b_n$ is divergent and $a_n \geq b_n$ for all $n$, then $\sum a_n$ diverges.

\textbf{Limit Comparison Test}:\\
Preconditions: $ \sum a_n$ and $\sum b_n$ are series with positive terms. \\
If $\lim_{n-> \infty} \frac{a_n}{b_n} = C$ where $C$ is finite and $C > 0$, then both series either converge or both diverge.

\subsection{Alternating Series}
$\sum_{n=1}^{\infty} (-1)^{n-1} b_n = b - b_2 + b_3 - b_4 + \ldots$ \\
If $b_{n+1} \leq b_n$ for all $n$ and $\lim_{n -> \infty} b_n = 0$ then the series converges.

\subsection{Absolute Convergence and the Ratio and Root Tests}
A series is \underline{absolutely convergent} if $\sum |a_n|$ is convergent. \\
A series is \underline{conditionally convergent} if it is convergent but not absolutely convergent. \\
If a series is absolutely convergent, then it is convergent. \\

\textbf{Ratio Test}:\\
If $\lim_{n -> \infty} |\frac{a_{n+1}}{a_n}| = L < 1$, then the series $\sum^{\infty}_{n=1} a_n$ is absolutely convergent.\\
If $\lim_{n -> \infty} |\frac{a_{n+1}}{a_n}| = L > 1$, then the series $\sum^{\infty}_{n=1} a_n$ is divergent.\\
If $\lim_{n -> \infty} |\frac{a_{n+1}}{a_n}| = 1$, then the Ratio Test is inconclusive.

\textbf{Root Test}:\\
If $\lim_{n -> \infty} \sqrt[n]{|a_n|} = L < 1$, then the series $\sum_{n=1}^{\infty} a_n$ is absolutely convergent.\\
If $\lim_{n -> \infty} \sqrt[n]{|a_n|} = L > 1$ or $lim_{n -> \infty} \sqrt[n]{|a_n|}$, then the series $\sum_{n=1}^{\infty} a_n$ is divergent.\\
If $\lim_{n -> \infty} \sqrt[n]{|a_n|} = 1$, then the Root Test is inconclusive.
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