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Define $$f_n(z)=\left(1+\frac{z}{n}\right)^n$$ Find the preimage of the closure of a square centered at $0$ with side length 1 under $f_n$. And explain in a geometric way that why $\lim_{n \to \infty}f_n(z)=e^z$.

My attempt:

Since $f_n$ maps a circle centered at $-n$ with radius $r$ to a circle centered at $0$ with radius $\left(\frac{r}{n}\right)^n$. So any circle centered at $-n$ with radius $r\leq \frac{n}{2^{1/n}}$ will maps inside the square.

However, I am confused how to find the preimage of the four corner parts of the square. I know the largest circle in the domain whose image cross with the square would be the circle centered at $-n$ with radius $r=\frac{n}{2^{1/2n}}$. By which they crossed at four vertexes of the square.

Thanks for any comments.

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