feat: reformat and add series

cheatsheets/ma1521/finals.tex

@@ -108,6 +108,7 @@
     \item $\lim\limits_{x\to c}\frac{tan(g(x))}{g(x)} = 1$
     \item $\lim\limits_{x\to 0}\frac{sin(x)}{x} = 1$
     \item $\lim\limits_{x\to 0}\frac{tan(x)}{x} = 1$
+    \item $\lim\limits_{n \to \infty} \frac{n!}{n^{n}} = 0$
   \end{itemize}
 
 
@@ -186,8 +187,6 @@ about x axis
     \item Vol Shell: $V = 2\pi\int^b_a x|f(x)-g(x)|dx$ (absolute!!)
     \item Length of curve: $\int^b_a \sqrt{1+f'(x)^2}dx$
   \end{itemize}
-\section{Series}
-TODO!!
 
   \section{Vectors}
   unit vector: $\hat{p} = \frac{p}{|p|}$, $\VV{AB} = \VV{OB} - \VV{OA}$
@@ -220,12 +219,11 @@ unit vector: $\hat{p} = \frac{p}{|p|}$, $\VV{AB} = \VV{OB} - \VV{OA}$
   \end{itemize}
 
   \subsection{Cross Product}
-$\vv{a} \times \vv{b} = \begin{vmatrix} 
-\vv{i} & \vv{j} & \vv{k} \\ 
-a_i & a_2 & a_3 \\
-b_i & b_2 & b_3
-\end{vmatrix} = \begin{pmatrix} (a_2b_3 - a_3b_2) \\ -(a_1b_3 - a_3b_1) \\ (a_1b_2 - a_2b_1) \end{pmatrix}$
-
+  $
+  \vv{a} \times \vv{b} =
+  \begin{vmatrix} \vv{i} & \vv{j} & \vv{k} \\ a_i & a_2 & a_3 \\ b_i & b_2 & b_3 \end{vmatrix} =
+  \begin{pmatrix} (a_2b_3 - a_3b_2) \\ -(a_1b_3 - a_3b_1) \\ (a_1b_2 - a_2b_1) \end{pmatrix}
+  $
   \begin{center}
     \begin{multicols}{2}
       $|\vv{a} \times \vv{b}| = |\vv{a}||\vv{b}|\sin\theta$
@@ -375,6 +373,77 @@ $S = \iint_R\sqrt{f_x^2 + f_y^2 + 1} dA$
    \subsection{Uranium Decay into Thorium}
    $U(t) = U_{0}e^{-k_ut}$, $k = \frac{\ln2}{\text{halflife}}, \frac{dU}{dt} = -k_{u}U$\\
    Thorium: $T(t) = \frac{K_{u}U_{0}}{K_{t}-K_{u}}(e^{-k_{u}t} - e^{-k_{t}t}), \frac{dT}{dt} = k_{u}U - k_{T}T$
+ \end{multicols*}
+ \newpage
+
+ \begin{multicols*}{4}
+   \setlength{\columnseprule}{0.25pt}
+   \setlength{\premulticols}{1pt}
+   \setlength{\postmulticols}{1pt}
+   \setlength{\multicolsep}{1pt}
+   \setlength{\columnsep}{2pt}
+
+   \section{Series}
+   \subsection{Geometric Series}
+   $\sum_{n=1}^{\infty}ar^{n-1}, a \ne 0$ converges to $\frac{a}{1-r}$ when $|r| < 1$, diverges otherwise
+
+   If series $\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\lim\limits_{n \to \infty} a_{n} = 0$
+
+   \subsection{Tests}
+   Decreasing function -> differentiate and see the range where $x < 0$
+   \begin{center}
+
+     \begin{tabular}{|>{\color{black}}p{0.2\linewidth} | >{\color{black}}p{0.7\linewidth}|}
+       \hline
+       Test & Method \\\hline
+       $n^{th}$ term & $\lim\limits_{n \to \infty} a_{n} \ne 0$ or does not exist, then divergent \\\hline
+       Integral & $f(n)=a_{n}$ is continuous, positive, decreasing function $\forall x\geq 1$ and $\int_{1}^{\infty}f(x)dx$ converges else divergent \\\hline
+       p-series & $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$convergent $\leftrightarrow p > 1$ \\\hline
+       Ratio & $0 \geq \lim\limits_{n \to \infty} |\frac{a_{n+1}}{a_{n}}|=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline
+       Root & $0 \geq \lim\limits_{n \to \infty} \sqrt[n]{a_{n}}=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline
+       Alternating series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline
+       Power Series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline
+     \end{tabular}
+ \end{center}
+
+ \subsection{Power Series}
+ $\sum_{n=0}^{\infty} c_{n}(x-a)^{n}$ converges at \textbf{ONE OF}
+ \begin{itemize}
+   \item $x=a$
+   \item For all $x$
+   \item converges if $|x-a| < R$ and diverges if $|x-a| > R$ (R is radius of convergence)
+ \end{itemize}
+
+ If $\lim_{n \to \infty} \left| \frac{c_{n+1}}{c_{n}} = L \right|$ or $\lim_{n \to \infty} \sqrt[n]{|c_{n}|}=L$, $L \in \mathbb{R}$ or $\infty$, then $R = \frac{1}{L}$
+
+ \subsection{Taylor and Maclaurin Series}
+ If f has power series repr @ $f = a$, $f(x) = \sum_{n=0}^{\infty} c_{n}(x-a)^{n}, |x-a| < R, R > 0$, then $c_{n} = \frac{f^{(n)}(a)}{n!}$. \\
+ \columnbreak
+ Maclaurin Series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{n}(0)}{n!}x^{n}$
+For $-\infty < x < \infty$
+\\ \;  % spacing
+\setlength\tabcolsep{1.5pt} % default value: 6pt
+\begin{tabular}{rl}
+           $\sin x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n+1)!} $
+        \\ $\cos x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^nx^{2n}}{(2n)!}$
+        \\ $e^x$ & $= \sum\limits^\infty_{n = 0} \frac{x^n}{n!}$
+\end{tabular}
+
+For $-1 < x < 1$
+\\ \;  % spacing
+\setlength\tabcolsep{1.5pt} % default value: 6pt
+\begin{tabular}{rl}
+           $\frac{1}{1 - x}$ & $= \sum\limits^\infty_{n = 0} x^n $
+        \\ $\frac{1}{1 + x}$ & $= \sum\limits^\infty_{n = 0} (-1)^nx^n $
+        \\ $\frac{1}{1 + x^2}$ & $= \sum\limits^\infty_{n = 0} (-1)^nx^{2n} $
+        \\ $\ln(1 + x)$ & $= \sum\limits^\infty_{n = 1} \frac{(-1)^{n - 1}x^n}{n} $
+        \\ $\tan^{-1}x$ & $= \sum\limits^\infty_{n = 0} \frac{(-1)^n}{2n + 1} x^{2n+1}$
+        \\ $\frac{1}{(1+x)^2}$ & $= \sum\limits^\infty_{n = 1} (-1)^{n-1}nx^{n-1}$
+        \\ $\frac{1}{(1-x)^2}$ & $= \sum\limits^\infty_{n = 1} nx^{n-1}$
+        \\ $\frac{1}{(1-x)^3}$ & $= \frac{1}{2} \sum\limits^\infty_{n = 2} n(n - 1)x^{n-2}$
+        \\ $(1 + x)^k$ & $= \sum\limits^\infty_{n = 0} \binom{k}{n}x^n$
+        \\   & $= 1 + kx + \frac{k(k-1)}{2!}x^2 + \dots$
+\end{tabular}
 
 
 \end{multicols*}