I am working through Naive Lie Theory by John Stillwell.
I can’t grasp why the following series converges:
$e^v = 1 + \frac{v}{1!} + \frac{v^2}{2!} + \frac{v^3}{3!} +$ ...
for all $v \in \mathbb{H}$, where $\mathbb{H}$ denotes the set of quaternions.
Stillwell’s proof is quite succinct:
For sufficiently large $n$, $\frac{|v|^n}{n!} < 2^{−n}$.
While I acknowledge this is valid for any series over $\mathbb{R}$ using the basic comparison test, on what basis can we even use these tests for quaternions? I find it pretty clear why one should be skeptical in generalizing these tests for convergence.