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The problem is exactly that of the title.

I attempted to apply Rouché's theorem, by setting another function $f(z)=e^z$ and comparing the modulus of $f(z)$ and $f(z)-P_n (z)$ at the circle $|z|=R$, but I failed to proceed anymore.
Is my approach correct? Or is there another solution?

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    See the answer by lhf to [this question](http://math.stackexchange.com/q/1769940) – 2017-01-15
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    Please include the question in the body of the post. – 2017-01-16
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    @PedroTamaroff Why is that necessary? – 2017-01-16
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    @zhw, it's good practice on Math.SE to [make the body of the post self-contained](http://meta.math.stackexchange.com/q/25695/5531). – 2017-01-17

2 Answers 2

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Let $R >0$. We know $\exp(z) \neq 0$ for all $z \in B_R(0)$. So, there is an $M>0$ such that $|\exp(z)| \geq M > 0$ for all $z \in B_R(0)$. Now, there exist $N \in \mathbb{N}$ such that for all $n \geq N$ we have $$ \frac{M}{2}>|\exp(z)-P_n(z)| \geq ||\exp(z)|-|P_n(z)||,$$ and thus $$ \frac{M}{2} > |\exp(z)|-|P_n(z)| > -\frac{M}{2}.$$

Therefore, $$ \frac{M}{2}+|\exp(z)| > |P_n(z)| > |\exp(z)|-\frac{M}{2}\geq \frac{M}{2} > 0$$ for all $n \geq N$.

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Suppose that exist $r>0 $such that $p_n$has zeros in $U:=\{|z|