(1) Let f : [0, too) - R be continuous and bounded. (a) Show that F ( s ) = / f(the - 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 E ( - 1 ) n An + 1 4V2 2 In(1 + V2) + 7 ) . (5 pts) n=0