## Summary of Alternating Series

### Essential Concepts

• For an alternating series $\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}$, if ${b}_{k+1}\le {b}_{k}$ for all $k$ and ${b}_{k}\to 0$ as $k\to \infty$, the alternating series converges.
• If $\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|$ converges, then $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges.

## Key Equations

• Alternating series

$\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\cdots \text{or}$

$\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}=\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\cdots$

## 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