Glossary

absolute convergence

if the series $\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|$ converges, the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ is said to converge absolutely

alternating series

a series of the form $\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}$ or $\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}$, where ${b}_{n}\ge 0$, is called an alternating series

alternating series test

for an alternating series of either form, if ${b}_{n+1}\le {b}_{n}$ for all integers $n\ge 1$ and ${b}_{n}\to 0$, then an alternating series converges

conditional convergence

if the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges, but the series $\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|$ diverges, the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ is said to converge conditionally