I'd like to start posting some of the questions I muse about on here to find out the thoughts of others and give other enthusiasts problems which are hopefully fun and nontrivial. My objective is to include some "hard" (to me) "open" (as far as I know) questions to counterbalance the many homework/project/reference questions we already get. Here is my first:
Recall that an arbitrary unit quaternion can be represented by $e^{p}$ where $p$ is a pure quaternion. Let $\{p_n\}_n$ be a sequence of pure quaternions and suppose further that $\sum_{n}^\infty p_n$ converges to $p$. Does it follow that $\prod_n^\infty e^{p_n}$ converges? Is there a (possibly trivial) $f$ such that it converges to $e^{f(p)}$?
If the word "quaternion" is dirty to you, you are free to replace it with "element of $SU(2)$".
Edit: The case of $q_n$ being a constant sequence is easy, has $f(p) = p$, and the convergence is "if and only if".