How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$
I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$
For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
Solution 1:
Presumably you know that $\lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)^n = e$. Try setting $n = \frac{k}{z}$ where $z$ is a constant and see where that gets you.