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% Cheatsheet for Series
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\title{Differential Equations Cheatsheet}
\author{}
\date{}

\begin{document}
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\begin{center}
  {\LARGE Series Cheatsheet}
\end{center}


%-------------------------------------------------------------------------------------------------------------
\section*{Definitions}

\subsection*{Basic Series}

{Infinite Sequence}:
$\left< s_n \right>$

\vspace{2mm}

{Limit/Convergence of a Sequence}:
$\lim_{n\rightarrow\infty} s_n = L $

\vspace{2mm}

{Infinite Serie}: (Partial sums)
$S_n$ = $\sum s_n = s_1 + s_2 + \cdots + s_n + \cdots $

\vspace{2mm}

{Geometric Serie}:
\[ \sum_{k=1}^{n} ar^{k-1} = S_n = a + ar + ar^2 + \dots + ar^{n-1} = \frac{a(1 - r^n)}{1-r} \]


\subsection*{Positive Series}

{Positive Serie}:
If all the terms $s_n$ are positive.

\vspace{2mm}

{Integral Test}:
If $f(n) = s_n$, continuous, positive, decreasing: $\sum s_n$ converges $\iff \int_1^{\infty} f(x)dx$ converges.

\vspace{2mm}

{Comparison Test}:
$\sum a_n$ and $\sum b_n$ where $a_k < b_k \quad (\forall k \geq m)$
\parbox{3in}{
\begin{enumerate}\tightitems
\item If $\sum b_n$ converges, so does $\sum a_n$
\item If $\sum a_n$ diverges, so does $\sum b_n$
\end{enumerate}
}

\vspace{2mm}

{Limit Comparison Test}:
$\sum a_n$ and $\sum b_n$ such that $\lim_{n\rightarrow\infty}
\frac{a_n}{b_n}$ exists,
$\sum a_n$ converges $\iff$ $\sum b_n$ converges.


\subsection*{Convergence}

{Alternating Serie}:
\[ \sum (-1)^{n+1} a_n = a_1 - a_2 + a_3 - a_4 + a_5 - \cdots \]

\vspace{2mm}

{Absolute Convergence}:
If $\sum |s_n|$ is convergent.

\vspace{2mm}

{Conditional Convergence}:
If $\sum s_n$ is convergent but \emph{not} absolutely convergent.

\vspace{2mm}

{Ratio Test}:
If $\lim_{n\rightarrow\infty} |\frac{s_{n+1}}{s_n}| = $
\parbox{2.5in}{
  \begin{itemize}\tightitems
  \item $< 1$: absolutely convergent
  \item $1$: (no conclusion)
  \item $>  1$ or $+\infty$: diverges
  \end{itemize}
}

{Root Test}:
If $\lim_{n\rightarrow\infty} \sqrt[n]{|s_n|} = $
\parbox{2.5in}{
\begin{itemize}\tightitems
\item $< 1$: absolutely convergent
\item $1$: (no conclusion)
\item $>  1$ or $+\infty$: diverges
\end{itemize}
}

{Uniform Convergence}:
If $\forall \epsilon > 0, \exists m$ such that
for each $x$ and every $n \geq m$,
\quad $ f_n(x) - f(x) < \epsilon $


\subsection*{Power Series}

{Power Serie}:
\[ \sum_{n=0}^{+\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots \]

\vspace{2mm}

{Power Serie About Zero}:
\[ \sum_{n=0}^{+\infty} a_nx^n = a_0 + a_1x + a_2x^2 + \cdots \]

\vspace{2mm}

{Taylor Serie} \\
If $f$ a function infinitely differentiable,
\[ \sum_{n=0}^{+\infty} \frac{f^{(n)}(c)}{n!} (x-c)^n \]

\vspace{2mm}

{MacLaurin Serie} \\
If $f$ a function infinitely differentiable,
\[ \sum_{n=0}^{+\infty} \frac{f^{(n)}(0)}{n!} x^n \]

\vspace{2mm}

{Taylor's Formula with Remainder} \\
$\exists x^*$ between $c$ and $x$ such that \\
\[ f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!} (x-c)^k + R_n(x) \]
\[ R_n(x) = \frac{f^{(n+1)}(x^*)}{(n+1)!} (x-c)^{n+1} \]


\subsection*{Applications}

{Application: Showing Function/Taylor-Series Equivalence}
\[ \lim_{n\rightarrow+\infty} R_n(x) = 0 \]

\vspace{2mm}

{Application: Approximating Functions or Integrals}
\[ R_n(x_0) < K \]

\vspace{2mm}

{Binomial Serie}
\[ (1+x)^r = 1 + \sum_{n=1}^{+\infty} \frac{r(r-1)(r-2)\cdots(r-n+1) }{n!} x^n \]



\subsection*{Common Series}

\begin{displaymath}
  e^x = \sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots
\end{displaymath}

\begin{displaymath}
  \dfrac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots +
\end{displaymath}
\begin{displaymath}
  \ln(1+x)= \sum_{n=0}^{\infty} (-1)^{n-1} \dfrac{x^n}{n} = x - \dfrac{1}{2}x^2 + \dfrac{1}{3}x^3 - \dfrac{1}{4}x^4 +
\end{displaymath}

\begin{displaymath}
  \sin x = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^{2n+1}}{(2n+1)!} =
  x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots
\end{displaymath}
\begin{displaymath}
  \cos x = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^{2n}}{(2n)!} =
  1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots
\end{displaymath}
\begin{displaymath}
  \sinh x = \sum_{n=0}^{\infty} \dfrac{x^{2n+1}}{(2n+1)!} =
  x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \dfrac{x^7}{7!} + \cdots
\end{displaymath}
\begin{displaymath}
  \cosh x = \sum_{n=0}^{\infty} \dfrac{x^{2n}}{(2n)!} =
  1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + \dfrac{x^6}{6!} + \cdots
\end{displaymath}



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% Copyright/distribution license.
{\vfill\hfill{\tiny Author: Martin Blais, 2009.
This work is licensed under the Creative Commons
``Attribution - Non-Commercial - Share-Alike'' license.}}
\end{document}
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