3. Let E(x) = L:~=o xk/k!.

(a) Prove that the series defining E(x) converges uniformly on any closed interval

[a,b].

(b) Prove that lb E(x) dx = E

(b) - E(a)

for all a,b E R.

(c) Prove that the function y = E(x) satisfies the initial value problem y' - y = 0, y(O) = 1.

(We shall see in Section 7.4 that E(x) = eX.)