6. The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { n } } { n - 1 }$ on its interval of convergence. (a) Find the interval of convergence for the Maclaurin series of $f$. Justify your answer. (b) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } - y = \frac { 4 x ^ { 2 } } { 1 + 2 x }$ for $| x | < R$, where $R$ is the radius of convergence from part (a).
WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM
, h ( t ) = h ( 0 ) + \int _ { 0 } ^ { t } g ( s ) d s = 0 + \int _ { 0 } ^ { t } 0 d s = 0$.
6. The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { n } } { n - 1 }$ on its interval of convergence.\\
(a) Find the interval of convergence for the Maclaurin series of $f$. Justify your answer.\\
(b) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } - y = \frac { 4 x ^ { 2 } } { 1 + 2 x }$ for $| x | < R$, where $R$ is the radius of convergence from part (a).
\section*{WRITE ALL WORK IN THE EXAM BOOKLET. \\
END OF EXAM}