If $\abs{f^{(n)}}\le M^n$ on $(c-\varepsilon,c+\varepsilon)$ for all $n$, the Taylor series of $f$ at $c$ converges to $f$ on that interval.Suggest a slogan
Let \(c\in\RR\), \(\varepsilon>0\), and \(f\in C^\infty((c-\varepsilon,c+\varepsilon))\) with values in \(\RR\). Suppose there exists \(M\geq0\) such that \[ \abs{f^{(n)}(x)} \leq M^n \quad\text{for all } n\geq0 \text{ and all } x\in(c-\varepsilon,c+\varepsilon). \] Then for every \(x\in(c-\varepsilon,c+\varepsilon)\), \[ f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(c)}{n!}(x-c)^n . \]