Example 7

Find an error estimate for the power series for ln((1 + x)/(1 - x)) valid for - c ≤ x ≤ c. Use it to approximate ln 2 within 0.00001.

From Example 5 we have the following error estimates for in (l + x) and - ln (1 - x) valid for - c ≤ x ≤ c.

We add the two sums and error estimates,

We wish to choose x so that (1 + x)/(l - x) = 2. Solving for x we get x = ⅓. Now set c = 5 and x = ⅓. The error estimate for x = ⅓ is

The error estimate drops below 0.00001 when m = 5. Thus ln 2 ~ 0.693146, error ≤ 0.00001.