Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$.
Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity
element and is power-associative, and then define $\:v^{\hspace{.02 in}n}\:$ in the obvious way.
Does it follow that for all vectors $\hspace{.02 in}v$, $\;\;\; \displaystyle\sum_{n=0}^{\infty} \; \left(\frac1{n!} \cdot v^{\hspace{.02 in}n}\hspace{-0.05 in}\right) \;\;\;$ exists?
If no, what if we additionally assume that $\hspace{.02 in}\beta\hspace{.02 in}$ is associative
and/or commutative and/or every vector has an inverse?