Prove that the power series for e^z does not converge uniformly on C.

[michael.wes]

Homework Statement

Prove that the power series for e^z does not converge uniformly on C.

Homework Equations

$$e^z=\sum_{k=0}^\infty z^k/k!$$

The Attempt at a Solution

The hint in the problem is to prove a proposition first:

If f_n is a sequence of entire functions that converges to 0 uniformly on C, then the f_n are eventually constant.

I proved this (an easy application of uniform convergence and Liouville's theorem), but I don't see how to use this to prove the main question.

Thanks for your help!
MW

 

[Dick]

Take the f_n to be the partial sums of the power series of e^z.