The exponential series converges uniformly on every bounded set by the Weierstrass $M$-test.Suggest a slogan
The series \(e^z=\sum_{n\ge0}\frac{z^n}{n!}\) converges uniformly on every bounded subset of \(\CC\), in particular on every disc in \(\CC\).
A bounded set lies in a disc \(\abs{z}\le R\) for some \(R>0\). There \(\abs{z^n/n!}\le R^n/n!\), and \(\sum_n R^n/n!=e^R<\infty\), so the Weierstrass \(M\)-test gives uniform convergence.