Please explain in detail and in legible handwriting, if applicable.

Power, Taylor and Maclaurin series and representation.

(d) Recall if a function f has a power series expansion at x = a, then its Taylor Series represen- tation is f (x ) = [ f (m) (a) (x - a)" = f0) ( a)+ f (a) f (2) (a ) n! (x - a)+ 2! (x - a) 2. f (3) (a) (x - a)3+... (1) n=0 1! 3! where f(n) _ d'f dyn. If f can be expanded at x = 0 and a = 0, the series is called a Maclaurin series, f(x) = _ f (") (02 xn = f(0) (0) + f() (0) f ( 2) (0 ) 2 1 f ( 3 ) (0 ) . as t .... (2) n=0 n! 1! 2! 3! Using Eq. 2 derive the Maclaurin representation of x2n+1 sin(x) = _ (-1) n=0 (2n + 1)