![\[ \begin{document} \section*{Series Convergence Tests} \begin{multicols}{2} \subsection*{Divergence Test} If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ diverges. \subsubsection*{Examples of Divergence Test} \begin{multicols}{2} \begin{enumerate} \item $\sum\limits_{n=1}^{\infty} n$ \item $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ \end{enumerate} \end{multicols} \subsection*{Comparison Test} Suppose $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are series with positive terms. If there exists a positive constant $C$ such that $a_n \leq Cb_n$ for all $n$, then: \begin{enumerate} \item If $\sum_{n=1}^{\infty} b_n$ converges, then $\sum_{n=1}^{\infty} a_n$ converges. \item If $\sum_{n=1}^{\infty} a_n$ diverges, then $\sum_{n=1}^{\infty} b_n$ diverges. \end{enumerate} \subsubsection*{Examples of Comparison Test} \begin{multicols}{2} \begin{enumerate} \item $\sum\limits_{n=1}^{\infty} \frac{1}{n^2 + 1}$ and $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$ \item $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ and $\sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ \end{enumerate} \end{multicols} \subsection*{Limit Comparison Test} Suppose $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are series with positive terms. If $\lim_{n \to \infty} \frac{a_n}{b_n}$ exists and is a positive finite number, then $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ either both converge or both diverge. \subsubsection*{Examples of Limit Comparison Test} \begin{multicols}{2} \begin{enumerate} \item $\sum_{n=1}^\infty \frac{1}{n^3+1}$ \item $\sum_{n=1}^\infty \frac{\sqrt{n}}{n^2+1} $ \end{enumerate} \end{multicols} \subsection*{Ratio Test} Suppose $\sum_{n=1}^{\infty} a_n$ is a series with positive terms. If $\lim_{n \to \infty} \frac{a_{n+1}}{a_n}$ exists and is a number $L < 1$, then $\sum_{n=1}^{\infty} a_n$ converges absolutely. If $\lim_{n \to \infty} \frac{a_{n+1}}{a_n}$ exists and is a number $L > 1$ or diverges, then $\sum_{n=1}^{\infty} a_n$ diverges. If $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 1$, then the test is inconclusive. \subsubsection*{Examples of Ratio Test} \begin{multicols}{2} \begin{enumerate} \item $\sum\limits_{n=1}^{\infty} \frac{2^n}{n!}$ \item $\sum\limits_{n=1}^{\infty} \frac{n^3}{3^n}$ \end{enumerate} \end{multicols} \subsection*{Root Test} Suppose $\sum_{n=1}^{\infty} a_n$ is a series with positive terms. If $\lim_{n \to \infty} \sqrt[n]{a_n}$ exists and is a number $L < 1$, then $\sum_{n=1}^{\infty} a_n$ converges absolutely. If $\lim_{n \to \infty} \sqrt[n]{a_n}$ exists and is a number $L > 1$ or diverges, then $\sum_{n=1}^{\infty} a_n$ diverges. If $\lim_{n \to \infty} \sqrt[n]{a_n} = 1$, then the test is inconclusive. \subsubsection*{Examples of Root Test} \begin{multicols}{2} \begin{enumerate} \item $\sum\limits_{n=1}^{\infty} \frac{3^n}{n!}$ \item $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$ \end{enumerate} \end{multicols} \subsection*{Integral Test} Let $f(x)$ be a continuous, positive, and decreasing function on $[1,\infty)$. Then the series $\sum_{n=1}^\infty f(n)$ converges if and only if the improper integral $\int_1^\infty f(x) dx$ converges. If the integral diverges, then so does the series. \subsubsection*{Examples of Integral Test} \begin{multicols}{2} \begin{enumerate} \item $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$ \item $\sum\limits_{n=2}^{\infty} \frac{1}{n\ln(n)}$ \end{enumerate} \end{multicols} \end{multicols} \end{document} \]](https://mlly6claceid.i.optimole.com/cb:P9zP.4d16a/w:1128/h:1015/q:mauto/f:best/ig:avif/https://learn4less.ca/wp-content/ql-cache/quicklatex.com-23bd88276c8693fd4afa86b364475324_l3.png "Rendered by QuickLaTeX.com")
Series Convergence Tests
Divergence Test
If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ diverges.
Examples of Divergence Test
- $\displaystyle\sum_{n=1}^{\infty} n$
- $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$
Comparison Test
Suppose $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are series with positive terms. If there exists a positive constant $C$ such that $a_n \leq Cb_n$ for all $n$, then:
- If $\sum_{n=1}^{\infty} b_n$ converges, then $\sum_{n=1}^{\infty} a_n$ converges.
- If $\sum_{n=1}^{\infty} a_n$ diverges, then $\sum_{n=1}^{\in