# Quaternions — Generalization of Taylor Series Polynomials to Higher Dimensional Numbers

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.

• 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. Aug 30 '20 at 22:34
• Basically it is because with quaternions we still have the rule $|v^n|=|v|^n$ for all natural numbers $n$. Sep 5 '20 at 19:01