I need help with part d. I want to see detailed steps.

As a refresher evaluate analytically the following expressions. Report your answer in simplest form. (a) Evaluate the respective sum and product. ( i ) E 2n 5 en (ii) II n = 0 2n + 2 n=1 n (b) Evaluate the following derivatives (Note. sinc(x) = sin(x) /x). (i) d dx -sinc(x) (ii)- a Inlex | dx (iii) d2 (c) Evaluate the following integrals. ( 1 ) xe * dx 2 TT (ii) .2 TT cos(x)dx +i sin(x)dx (d) Recall if a function f has a power series expansion at x = a, then its Taylor Series represen- tation is f(x) = [ f (n) (a) ( x - a) " = f (0) ( a ) + f ( 1 ) ( a ) , 1 ! ( x - a ) + f ( 2 ) ( a ) n=0 n! 2 ! ( x - a ) 2 + f ( 3 ) ( a ) 3 ! ( x - a ) 3 + ... ( 1 ) an f where f() = an. If f can be expanded at x = 0 and a = 0, the series is called a Maclaurin series, f (x) = [ f (n) (0) xh = f(0 ) (0 ) + f() (0 ) | f ( 2 ) (0 ) 12 + f (3) (0 ) n! + .... 3! (2) n=0 1! 2! Using Eq. 2 derive the Maclaurin representation of sin(x) = E(-1)n_x2n+1 (2n + 1)