The Taylor series for a function $f$ about $x = 1$ is given by $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n } ( x - 1 ) ^ { n }$ and converges to $f ( x )$ for $| x - 1 | < R$, where $R$ is the radius of convergence of the Taylor series. (a) Find the value of $R$. (b) Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 1$. (c) The Taylor series for $f ^ { \prime }$ about $x = 1$, found in part (b), is a geometric series. Find the function $f ^ { \prime }$ to which the series converges for $| x - 1 | < R$. Use this function to determine $f$ for $| x - 1 | < R$.
The Taylor series for a function $f$ about $x = 1$ is given by $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n } ( x - 1 ) ^ { n }$ and converges to $f ( x )$ for $| x - 1 | < R$, where $R$ is the radius of convergence of the Taylor series.\\
(a) Find the value of $R$.\\
(b) Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 1$.\\
(c) The Taylor series for $f ^ { \prime }$ about $x = 1$, found in part (b), is a geometric series. Find the function $f ^ { \prime }$ to which the series converges for $| x - 1 | < R$. Use this function to determine $f$ for $| x - 1 | < R$.