Example 2: arctan

Approximate arctan ½ within 0.001. The power series for arctan x is

r = 1. Setting x = ½,

This an an alternating series. The last term is less than 0.001,

Therefore

error ≤ 0.0002.

Adding up,

arctan ½ ~ 0.4635, error ≤ 0.0002.

The series

r = 1

can be used to approximate π. We start with

Setting x = 1/√3 in the series,

or

This is an alternating series, so

, error ≤

~ 0.9072, error ≤ 0.0004.

Dividing everything by √3/6 we get

π ~ 3.1426, error ≤ 0.0013.