Example 5.11. Estimate the error made when replacing the sum of the series

$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots
$$

with the sum of its first four terms.
Solution. The given alternating series converges (see problem 5.10). The error $\Delta S_{4}$, obtained by replacing the sum $S$ of this series with the sum of its first four terms, is less than the absolute value of the fifth term: $\Delta S_{4}<0.2$.

## 5.4. Absolute and Conditional Convergence of an Alternating Series

A series

$$
a_{1}+a_{2}+\ldots+a_{n}+\ldots
$$

is called alternating if among its terms there are both positive and negative numbers.

Convergence criterion for an alternating series. If the series

$$
\left|a_{1}\right|+\left|a_{2}\right|+\ldots+\left|a_{n}\right|+\ldots
$$

composed of the absolute values of the terms of series (1), converges, then the series (1) also converges.

An alternating series (1) is called absolutely convergent if the series (2), composed of the absolute values of the terms of the given series (1), converges.

A convergent alternating series is called conditionally convergent if the series, composed of the absolute values of its terms, diverges.