$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$
This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
2026-04-07 22:59:21.1775602761
For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane
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It means that if you want to compute $\exp(z)$ to a desired precision, I can tell you how many summands will suffice without knowing $z$; provided you tell me that $|z|$ does not exceed a certain threshold (that is, my answer will depend only on that threshold and the desired precision, not on the exact value of $z$).