Example 5

Given a constant c where -1 < c < 1, find a simple error estimate for the power series

valid for - 1 < x ≤ c. We start with the equation

(1)

For -1 < t ≤ c we have

1 - t ≥ 1 - c,

Integrating Equation 1 from 0 to x we have (2)

and

Multiplying Equation 2 by - 1 and setting m = n + 1 we have the following error estimate for ln (1 - x), valid for - 1 < x ≤ c.

(3)