Suppose $P_n(z)=1+z/1!+z^2/2!+\cdots+z^n/n!$ and $Q_n(z)=P_n(z)-1$ where $n=1,2,3,\cdots$. What can you say about the location of the zeros of $P_n$ and $Q_n$ for large $n$?

It is an exercise in Rudin's Real and Complex Analysis (Chapter 10).

I want to estimate $P'/P$ and use principle argument. For example, for large $n$ enough, there exist circles $C(0,r)=\{z\in\mathbb{C}: |z|=r\}$ and $C(0,R)$ s.t., $$\frac{1}{2\pi i}\int_{C(0,r)}\frac{P'(z)}{P(z)}\,dz<1.$$ and $$\frac{1}{2\pi i}\int_{C(0,R)}\frac{P'(z)}{P(z)}\,dz>n.$$ But it seems difficult for the polynomials $P$ and $Q$. Could anybody give me a hint?

  • $\begingroup$ For $P$, look at answers of this question. $\endgroup$ – achille hui Oct 20 '16 at 8:23
  • $\begingroup$ @achillehui Thank you. M. Zemyan' paper may be useful. $\endgroup$ – Edelweiss Ntu Oct 20 '16 at 8:42

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