Glossary

comparison test

if $0\le {a}_{n}\le {b}_{n}$ for all $n\ge N$ and $\displaystyle\sum _{n=1}^{\infty }{b}_{n}$ converges, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges; if ${a}_{n}\ge {b}_{n}\ge 0$ for all $n\ge N$ and $\displaystyle\sum _{n=1}^{\infty }{b}_{n}$ diverges, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ diverges

limit comparison test

suppose ${a}_{n},{b}_{n}\ge 0$ for all $n\ge 1$. If $\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to L\ne 0$, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ and $\displaystyle\sum _{n=1}^{\infty }{b}_{n}$ both converge or both diverge; if $\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to 0$ and $\displaystyle\sum _{n=1}^{\infty }{b}_{n}$ converges, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges. If $\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to \infty$, and $\displaystyle\sum _{n=1}^{\infty }{b}_{n}$ diverges, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ diverges