NUNLLLLU (1) Let f : [0, +oo) - RR be continuous and bounded. (a) Show that F (s ) = / f (t) e - stat is defined for all s > 0. (b) Consider an arbitrary so > 0. Prove that F is represented by a power series F(s) = Enzo an(s - so)" which has a radius of convergence R 2 80. (5 pts) (2) Prove the indentity (2n - 1)! n = 0 (2n + 1) (2n)!' where we use the convention (-1)!! = 0!! = 1. (5 pts) (3) Prove the indentity ( - 1 ) n 1 An + 1 4V2 (2 In (1 + V2) + +). (5 pts) n= 0 (4) Prove that the series 00 f (2) = > cos(nx) n= 1 n converges pointwise on the open interval (0, 27) and uniformly on any closed subinterval [a, b] c (0, 27). Find a closed form expression for f (x). (5 pts)