Series Tests

If $a_{n}$ ...	then try ...
does not approach zero	Test for Divergence
has factorials or exponentials $\left(e^{n}\right)$	Ratio Test
alternates signs; has $\left(-1\right)^{n}$	Alternating Series Test, then test $\textstyle \sum \left\|a_{n}\right\|$ for absolute convergence.
involves fractions with powers of $n$ , like $n^{2}$ , $n^{3}$ , $n^{4}$ , etc.	Limit Comparison Test or Straight Comparison.
comes from a decreasing, easy-to-integrate function	Integral Test

Test for Divergence

$\lim _{n\to \infty }a_{n}{\begin{cases}\neq 0&{\mbox{divergent}}\\=0&{\mbox{unknown}}\end{cases}}$

$\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|{\begin{cases}<1&{\mbox{absolutely convergent}}\\=1&{\mbox{unknown}}\end{cases}}$

$\lim _{n\to \infty }\left|a_{n}\right|{\begin{cases}=0&{\mbox{convergent}}\\\neq 0&{\mbox{divergent}}\end{cases}}$

$a_{n}<b_{n}{\begin{cases}b_{n}\ {\mbox{is convergent}}&a_{n}\ {\mbox{is convergent}}\\b_{n}\ {\mbox{is divergent}}&{\mbox{unknown}}\end{cases}}$

$a_{n}>b_{n}{\begin{cases}b_{n}\ {\mbox{is convergent}}&{\mbox{unknown}}\\b_{n}\ {\mbox{is divergent}}&a_{n}\ {\mbox{is divergent}}\end{cases}}$

$\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}{\begin{cases}>0&{\mbox{both converge or diverge}}\\=0&{\mbox{unknown}}\end{cases}}$

$\sum _{n=b}^{\infty }{a_{n}}$

$\left({\mbox{let}}\ f\left(x\right)=a_{n}\ {\mbox{for all}}\ x=n\right)$

$\quad \lim _{t\to \infty }\int _{b}^{t}f\left(x\right)\;dx\ {\mbox{converges}}\rightarrow \sum \ {\mbox{converges}}$