The function $f$ is defined by the power series $f ( x ) = x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 } - \frac { x ^ { 7 } } { 7 } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n + 1 } } { 2 n + 1 } + \cdots$ for all real numbers $x$ for which the series converges. (a) Using the ratio test, find the interval of convergence of the power series for $f$. Justify your answer. (b) Show that $\left| f \left( \frac { 1 } { 2 } \right) - \frac { 1 } { 2 } \right| < \frac { 1 } { 10 }$. Justify your answer. (c) Write the first four nonzero terms and the general term for an infinite series that represents $f ^ { \prime } ( x )$. (d) Use the result from part (c) to find the value of $f ^ { \prime } \left( \frac { 1 } { 6 } \right)$.
The function $f$ is defined by the power series $f ( x ) = x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 } - \frac { x ^ { 7 } } { 7 } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n + 1 } } { 2 n + 1 } + \cdots$ for all real numbers $x$ for which the series converges.\\
(a) Using the ratio test, find the interval of convergence of the power series for $f$. Justify your answer.\\
(b) Show that $\left| f \left( \frac { 1 } { 2 } \right) - \frac { 1 } { 2 } \right| < \frac { 1 } { 10 }$. Justify your answer.\\
(c) Write the first four nonzero terms and the general term for an infinite series that represents $f ^ { \prime } ( x )$.\\
(d) Use the result from part (c) to find the value of $f ^ { \prime } \left( \frac { 1 } { 6 } \right)$.