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Series : Uniform Convergence and No Zeros

Fix R>0. Show that, if n is large enough, then P_n(z)=1+z+z^2/2!+z^3/3!+...+z^n/n! has no zeros in {z:|z|<=R}

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Proof:
We know that P_n(z)->exp(z) as n->oo. Especially, in the region
D={z:|z|<=R}, P_n(z) converges to exp(z) uniformly.
Now for ...

Solution Summary

It is shown that, if n is large enough, then P_n(z)=1+z+z^2/2!+z^3/3!+...+z^n/n! has no zeros in {z:|z|<=R} (R>0). The response was given a rating of "5/5" by the student who originally posted the question.

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