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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.

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    $\begingroup$ The quaternions are a finite-dimensional normed vector space (en.wikipedia.org/wiki/Normed_vector_space) and the usual development of convergent sequences etc. can be done in this setting with basically no modifications. $\endgroup$ Aug 30 '20 at 22:34
  • $\begingroup$ Basically it is because with quaternions we still have the rule $|v^n|=|v|^n$ for all natural numbers $n$. $\endgroup$ Sep 5 '20 at 19:01

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