15. Compute ln 2

We will use the formula

If we let x = 2, then x - 1 = 1 and we get

This will converge to ln 2. However, it converges very slowly. It is quicker to use the fact that

ln 2 = - ln(1/2)

and if we use

x = 1/2

then

x - 1 = -1/2

so using the formula above

taking the sum out one more term. Note that the negative outside the parentheses will turn all these terms positive. Converting to decimals we get

adding in the next term again. We see that we are getting closer and closer to the value we get of

.6931471806. . .

that we get from a calculator.