I asked essentially this over two weeks ago [on MSE](https://math.stackexchange.com/questions/513016/how-general-is-the-convergence-of-the-exponential-functions-power-series), and nothing was else was posted to that question.
<br><br><br>
Let $\mathbf{V}$ be a [Fréchet space](http://en.wikipedia.org/wiki/Frechet_space) whose underlying set is $V$.
<br>
Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map
<br>
that has an identity element and is [power-associative](http://en.wikipedia.org/wiki/Power-associative).
<br>
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
<br>
and/or commutative and/or every vector has an inverse?
<br><br>