Chapter 11, Problem 65RE

a.

To determine

To find: The Taylor series for $\ln \left(1+x\right)$ at $x=1$.

b.

To determine

To find: The Taylor series for $\ln \left(1-x\right)$ by using the identity $\ln 2=-\ln \frac{1}{2}$.

c.

To determine

To find: The Taylor series for $f\left(x\right)=\ln \left(\frac{1+x}{1-x}\right)$ by the use of the property $\ln \left(\frac{a}{b}\right)=\ln a-\ln b$.

d.

To determine

To find: The value of x for which the series in part (c) is approximate to $\ln 2$ and the resulting result for $\ln 2$.

e.

To determine

To explain: The reason for the three series derived in parts (a)-(d) gives the best approximation to $\ln 2$.