Prove that the series
\[ f ( x ) = \sum _ { n = 1 } ^ { \infty } { \frac { \sin n x } { n ^ { 5 / 2 } } } \]
converges for all \(x \in \mathbf { R }\) and that \(f ( x )\) is a continuous function on \(\mathbf { R }\) with a continuous derivative. State clearly any facts you assume.