Alternating Series

analysis

Definition

Alternating Series

An infinite series ${\sum }_{n=0}^{\infty }{a}_n$ is called alternating if its terms alternate in sign. This is typically written in the form:

$$\sum _{n=0}^{\infty }(-1{)}^n{b}_n\text{or}\sum _{n=0}^{\infty }(-1{)}^{n+1}{b}_n$$

where $b_n>0$ for all $n$.

Convergence

An alternating series does not necessarily converge. The most common tool for testing the convergence of such series is the Leibniz Criterion.

Leibniz's Theorem

The alternating series ${\sum }_{n=0}^{\infty }(-1{)}^n{b}_n$ converges if:

1. The sequence $(b_n)$ is monotonically decreasing ($b_{n+1}\le b_n$).
2. The sequence $(b_n)$ is a zero sequence (${\lim }_{n\to \infty }{b}_n=0$).

Examples

• Alternating Harmonic Series: ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$. This series is conditionally convergent.
• Geometric Alternating Series: ${\sum }_{n=0}^{\infty }(-1{)}^n{\left(\frac{1}{2}\right)}^n=1-\frac{1}{2}+\frac{1}{4}-\cdots =\frac{2}{3}$. This series is absolutely convergent.