I asked essentially this over two weeks ago on MSE, and nothing was else was posted to that question.


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.
For vectors $v$ and non-negative integers $n$, define $\hspace{.02 in}v^{\hspace{.02 in}n}\hspace{.02 in}$ in the obvious way.

Does it follow that for all vectors $\hspace{.02 in}v$, $\;\;\; \displaystyle\sum_{n=0}^{\infty} \; \left(\hspace{-0.03 in}\frac1{n!}\hspace{-0.05 in}\cdot \hspace{-0.02 in}v^{\hspace{.02 in}n}\hspace{-0.05 in}\right) \;\;\;$ 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?