The function $f$ has a Taylor series about $x = 1$ that converges to $f ( x )$ for all $x$ in the interval of convergence. It is known that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - \frac { 1 } { 2 }$, and the $n$th derivative of $f$ at $x = 1$ is given by $f ^ { ( n ) } ( 1 ) = ( - 1 ) ^ { n } \frac { ( n - 1 ) ! } { 2 ^ { n } }$ for $n \geq 2$. (a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$. (b) The Taylor series for $f$ about $x = 1$ has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer. (c) The Taylor series for $f$ about $x = 1$ can be used to represent $f ( 1.2 )$ as an alternating series. Use the first three nonzero terms of the alternating series to approximate $f ( 1.2 )$. (d) Show that the approximation found in part (c) is within 0.001 of the exact value of $f ( 1.2 )$.
The function $f$ has a Taylor series about $x = 1$ that converges to $f ( x )$ for all $x$ in the interval of convergence. It is known that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - \frac { 1 } { 2 }$, and the $n$th derivative of $f$ at $x = 1$ is given by $f ^ { ( n ) } ( 1 ) = ( - 1 ) ^ { n } \frac { ( n - 1 ) ! } { 2 ^ { n } }$ for $n \geq 2$.\\
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$.\\
(b) The Taylor series for $f$ about $x = 1$ has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer.\\
(c) The Taylor series for $f$ about $x = 1$ can be used to represent $f ( 1.2 )$ as an alternating series. Use the first three nonzero terms of the alternating series to approximate $f ( 1.2 )$.\\
(d) Show that the approximation found in part (c) is within 0.001 of the exact value of $f ( 1.2 )$.