Example 6

Use Example 5 to approximate ln ½ within 0.01. We set c = x = ½ in Equation 3.

Table 9.9.1 shows approximate values and error estimates.

We see that the error estimate drops below 0.01 when m = 5. So

ln ½ ~ -0.689, error ≤ 0.01.

Since ln ½ = -ln 2, we also have

ln 2 ~ 0.689, error ≤ 0.01.

A more rapidly converging series for ln 2 can be obtained in the following way. Any number a > 1 can be put in the form

, 0 < x < 1.

We simply take

By the rules of logarithms,

We can subtract two series by the Sum Rule, whence

, r = 1,

, r = 1,

, r = 1.

This power series is convenient because half of the terms are zero.