Approximating In 2 Consider the following three ways to

Chapter 1, Problem 49RE

(choose chapter or problem)

Approximating ln 2  Consider the following three ways to approximate ln 2.

a. Use the Taylor series for ln (1+x) centered at 0 and evaluate it a x = 1 (convergence was asserted in Table 9.5). Write the resulting infinite series.

b. Use the Taylor series for ln (1 - x) centered at 0 and the identity \(\ln 2=-\ln\frac{1}{2}\). Write the resulting infinite series.

c. use the property ln (a/b) = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for \(f(x)=\ln (\frac{1+x}{1-x})\) centered at 0.

d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)-(d) gives the best approximation to ln 2? Which series gives the worst approximation? Can you explain why?