Let E(x) = k=0 xk/k!. a) Prove that the series defining E(x) converges uniformly on any closed

Let E(x) = ˆ‘ˆžk=0 xk/k!.
a) Prove that the series defining E(x) converges uniformly on any closed interval [a, b].
b) Prove that

for all a, b ˆˆ R.
c) Prove that the function y = E(x) satisfies the initial value problem
y'-y = 0, y(0 ) = l.
[We shall see in Section 7.4 that E(x) = ex.]

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la E(x) dx = E(b)-E(a)