The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Alternating Series Test


Power series are one of the most important methods of approximation in mathematics. Consider a power series

f(x) = a0 + a1x + a2x2 + ... + anxn + ....

The partial sums give approximate values for the function,

f(x) ~ a0 + axx + a2x2 + ... + anxn,

and the tails En give the error in the approximation,

f(x) = a0 + a1x + a2x2 + ... + anxn + En.

If we can estimate the error En we can compute approximate values for f(x) to any desired degree of accuracy.

In this section we shall give two simple methods of estimating the error. A more general method will be given in the next section. Our first method is to use the Alternating Series Test. It can be applied whenever a power series is alternating.

Example 1: ln
Example 2: arctan
Example 3: e

Last Update: 2006-11-08