(Requires calculus) This exercise shows how generating
Chapter 10, Problem 31E(choose chapter or problem)
Problem 31E
(Requires calculus) This exercise shows how generating functions can be used to solve the recurrence relation (n +1)an+1 = an + (1 /n!) for n ≥ 0 with initial condition a0=1.
a) Let G(x) be the generating function for {an}. Show that G'(x) = G(x) + ex and G(0) = 1.
b) Showfrompart (a) that (e− xG(x))' = 1, andconclude that G(x) = xex + ex.
c) Use part (b) to find a closed form for an.
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