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% Document Info
\author{ebabor}
\pdfinfo{
  /Title (sequences-and-series.pdf)
  /Creator (Cheatography)
  /Author (ebabor)
  /Subject (Sequences and Series Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Sequences and Series Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{ebabor} via \textcolor{DarkBackground}{\uline{cheatography.com/69500/cs/17544/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}ebabor \\
  \uline{cheatography.com/ebabor} \\
  \end{tabulary}
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\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 22nd October, 2018.\\
   Updated 22nd October, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
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\begin{document}
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{General Rules}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Telescoping and Geometric series are the only types of series that you can estimate sums from. So, you must use these test's properties to estimate these sums} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If the question is asking for absolute convergence or conditional convergence. You will need to use the Ratio Test, Root Test, or the definition of Absolute/Conditional Convergence} \tn 
% Row Count 8 (+ 4)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Must show {\bf{ALL}} work to receive full credit for questions. Study the process to solve the problems, don't just guess through the review} \tn 
% Row Count 11 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.32733 cm} x{1.41887 cm} x{1.8308 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Tests}}  \tn
% Row 0
\SetRowColor{LightBackground}
Test for \seqsplit{Divergence(TFD)} & \seqsplit{Inconclusive} & You absolutely cannot determine if a series is convergent from this test. \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
 & Diverges & If limit of series ≠0 or ∞ \tn 
% Row Count 7 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Integral Test & Converges & If integral of series \textless{}∞ \tn 
% Row Count 9 (+ 2)
% Row 3
\SetRowColor{white}
 & Diverges & If integral of series =∞ \tn 
% Row Count 11 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Ratio Test & \seqsplit{Converges/Converges} Absolutely & If limit of 0≤|(a`k+1`) /(a`k`)|\textless{}1 \tn 
% Row Count 14 (+ 3)
% Row 5
\SetRowColor{white}
 & Diverges & If limit \textgreater{}1 \tn 
% Row Count 15 (+ 1)
% Row 6
\SetRowColor{LightBackground}
 & \seqsplit{Inconclusive} & If limit =1 \tn 
% Row Count 16 (+ 1)
% Row 7
\SetRowColor{white}
Root Test & \seqsplit{Converges/Converges} Absolutely & If 0≤limit of the k\textasciicircum{}th\textasciicircum{} root of |a`k`|\textless{}1 \tn 
% Row Count 19 (+ 3)
% Row 8
\SetRowColor{LightBackground}
 & Diverges & If limit of the k\textasciicircum{}th\textasciicircum{} root of |a`k`|\textgreater{}1 \tn 
% Row Count 22 (+ 3)
% Row 9
\SetRowColor{white}
 & \seqsplit{Inconclusive} & If limit of k\textasciicircum{}th\textasciicircum{} root of |a`k`|=1 \tn 
% Row Count 25 (+ 3)
% Row 10
\SetRowColor{LightBackground}
Direct Comparison Test(CT) & Converges & If ∑b`k` converges AND b`k` is the larger of the two functions \tn 
% Row Count 29 (+ 4)
% Row 11
\SetRowColor{white}
 & Diverges & If ∑b`k` diverges AND b`k` is smaller of two functions \tn 
% Row Count 33 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{x{1.32733 cm} x{1.41887 cm} x{1.8308 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Tests (cont)}}  \tn
% Row 12
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Limit Comparison Test(LCT) & Converges & If b`k` converges AND limit of 0\textless{}(a`k`)/(b`k`)\textless{}∞ \tn 
% Row Count 4 (+ 4)
% Row 13
\SetRowColor{white}
 & Diverges & If b`k` AND limit of 0\textless{}(a`k`)/(b`k`)\textless{}∞ \tn 
% Row Count 7 (+ 3)
% Row 14
\SetRowColor{LightBackground}
\seqsplit{Alternating} Series Test(AST) & Converges & If all 3 conditions for AST are met \tn 
% Row Count 10 (+ 3)
% Row 15
\SetRowColor{white}
 & Diverges & If limit condition fails, ∑a`k` is immediately divergent by TFD \tn 
% Row Count 15 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.28156 cm} x{1.19002 cm} x{2.10542 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Properties of Special Series}}  \tn
% Row 0
\SetRowColor{LightBackground}
Geometric Series & Converges & If Absolute Value of r\textless{}1. Converges at S=(first term)/(1-r) \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
 & Diverges & If Absolute Value of r≥1 \tn 
% Row Count 6 (+ 2)
% Row 2
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P-Series & Converges & If p\textgreater{}1 \tn 
% Row Count 7 (+ 1)
% Row 3
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 & Diverges & If p≤1 \tn 
% Row Count 8 (+ 1)
% Row 4
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\seqsplit{Telescoping} & Converges & Value that the limit of the remaining terms approach \tn 
% Row Count 11 (+ 3)
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 & Diverges & Almost never. On the test it will converge \tn 
% Row Count 14 (+ 3)
% Row 6
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Definition of \seqsplit{Convergence} & Absolute \seqsplit{Convergence} & If and only if ∑|a`k`| is convergent \tn 
% Row Count 17 (+ 3)
% Row 7
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 & \seqsplit{Conditional} \seqsplit{Convergence} & If and only if ∑a`k` is convergent, but ∑|a`k`| is divergent \tn 
% Row Count 21 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.04057 cm} x{2.93643 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{When to Use Tests}}  \tn
% Row 0
\SetRowColor{LightBackground}
Properties & If you can identify the series as a geometric, p, or telescoping series, then use their respective properties. If the given series looks {\emph{close}} to one of these series {\bf{see if you can use algebra to rearrange it into one of them}} \tn 
% Row Count 11 (+ 11)
% Row 1
\SetRowColor{white}
Test for Divergence(TFD) & Should at least eyeball this test first to see if the limit of the series does not approach 0. If series does not approach 0, then {\bf{∑a`k` divergent by TFD}} \tn 
% Row Count 18 (+ 7)
% Row 2
\SetRowColor{LightBackground}
Comparison Tests(CT and LCT) & {\bf{ONLY POSITIVE TERMS!}} If you can tell if the series has negative terms,((-1)\textasciicircum{}k\textasciicircum{} or sin/cos), do not use this test. If series has is rational and has a root in the denominator, compare with a p-series. |a`k`| gives use absolute convergence \tn 
% Row Count 29 (+ 11)
% Row 3
\SetRowColor{white}
Alternating Series Test(AST) & Series with (-1)\textasciicircum{}k\textasciicircum{} can be testes with AST \tn 
% Row Count 31 (+ 2)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{x{2.04057 cm} x{2.93643 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{When to Use Tests (cont)}}  \tn
% Row 4
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Integral Test & {\bf{ONLY POSITIVE TERMS!}} If you can look at the function and easily take the integral, it is probably good to use this test \tn 
% Row Count 6 (+ 6)
% Row 5
\SetRowColor{white}
Ratio Test & If series contains: k!, or powers {\emph{and}} exponentials, {\bf{almost guaranteed to use ratio test}} \tn 
% Row Count 11 (+ 5)
% Row 6
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Root Test & If the entire series can be written to the k\textasciicircum{}th\textasciicircum{} power, you can use the root test \tn 
% Row Count 15 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Integral Test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Conditions: \newline % Row Count 1 (+ 1)
1. f(x) is positive on its interval \newline % Row Count 2 (+ 1)
2. f(x) is continuous on its interval \newline % Row Count 3 (+ 1)
3. f(x) decreasing as x-\textgreater{}∞ (derivative is negative) \newline % Row Count 5 (+ 2)
* Must change a`k` to a function in order to take derivative \newline % Row Count 7 (+ 2)
* Integral starts off from k to ∞, so you must change the integral to k to t with limit as t-\textgreater{}∞ \newline % Row Count 10 (+ 3)
* Answer you get is not where the ∑a`k` converges% Row Count 12 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Alternating Series Test}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Conditions: \newline % Row Count 1 (+ 1)
1. b`k`\textgreater{}0 \newline % Row Count 2 (+ 1)
2. b`k`≥b`k+1` \newline % Row Count 3 (+ 1)
3. limit of b`k`=0 \newline % Row Count 4 (+ 1)
* If ∑b`k` fits all three conditions, ∑a`k` convergent by AST \newline % Row Count 6 (+ 2)
* If 3\textasciicircum{}rd\textasciicircum{} condition fails, ∑a`k` is divergent by TFD \newline % Row Count 8 (+ 2)
* If series contains (-m)\textasciicircum{}k\textasciicircum{}, pull (-1)\textasciicircum{}k\textasciicircum{} out and keep m\textasciicircum{}k\textasciicircum{} in b`k`% Row Count 10 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{0.9154 cm} x{2.15119 cm} x{1.51041 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Integral Remainder}}  \tn
% Row 0
\SetRowColor{LightBackground}
Known n\textasciicircum{}th\textasciicircum{} Value & Solving for S`n`(sum of the series approximation) & Plug n into the series \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
 & Solving for R`n` (approximation of the remainder) & Solve integral from n to ∞ \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Known Error Bound & Set the integral of f(x) from n to ∞ less than error bound. Once solved, answer will be given in terms of n\textless{}\#. Must round up the number since series only use integers and if you rounded down, the value of the integral  would be larger than the error bound &  \tn 
% Row Count 21 (+ 15)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.05271 cm} x{2.47158 cm} x{1.05271 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Alternating Series Remainder}}  \tn
% Row 0
\SetRowColor{LightBackground}
Known Error Bound & Set error bound less than b`n+1`. Solve for a \#\textgreater{}n, round up n to next highest whole number &  \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
 & If the inequality is very difficult to solve, the use of a table, shown below, is acceptable. When the middle column, b`n+1`, is less than third column, error bound, then that value is you final answer for n. Since the original variable in the equation is k, and k=n+1, then the value of the final term you can stop on to reach your error bound will be k &  \tn 
% Row Count 22 (+ 17)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{} \tn 
% Row Count 22 (+ 0)
% Row 3
\SetRowColor{white}
n\textasciicircum{}th\textasciicircum{} term\{\{br\}\} & b`n+1`\{\{br\}\} & Error Bound\{\{br\}\} \tn 
% Row Count 24 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}