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For $n=1,2,\cdots, $ let $P_n(z) = \sum_{k=0}^{n} \frac{z^k}{k!}$ where $z \in\mathbb{C}$. I want to show that for every $R>0$, there is a positive integer $N$ such that for all $n\geq N$ and $|z|<R$, $P_n(z) \neq 0$.

I am confused this theorem with the "fundamental theorem of algebra"

Every non-constant polynomial has at least one zero in $\mathbb{C}$. i.e., Let $p(z) = a_0 + a_1 z + \cdots + a_n z^n$, $a_i \in \mathbb{C}$, $n\geq 1$, $a_n \neq 0$. Then $\exists z_0 \in \mathbb{C}$ such that $p(z_0)=0$.

How one can prove the above theorem? (Not the fundamental theorem of algebra)

phy_math
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1 Answers1

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Fix $R>0$. First recall that $P_n(z) \to \exp(z)$ uniformly on $K_R:=\{z \in \mathbb{C}:|z| \le R\}$. Assume by contradiction that there are sequences $n_1 < n_2 < n_3 < \dots$ in $\mathbb{N}$ and $z_1,z_2,z_3, \dots$ in $K_R$ with $P_{n_k}(z_k)=0$. Since $K_R$ is compact w.l.o.g (by choosing a subsequence) let $(z_k)$ be convergent with limit $w$, say. Now, as $k \to \infty$ we have $$ |P_{n_k}(z_k)-\exp(z_k)| \to 0 $$ by uniform convergence on $K_R$, and $$ |P_{n_k}(z_k)-\exp(z_k)| = |\exp(z_k)| \to |\exp(w)|. $$ Thus $\exp(w)= 0$, a contradiction. Of course, by the fundamental theorem of algebra each $P_n$ has zeros but with $n$ the zeros are wandering to infinity.

Gerd
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