@DanieleTampieri thank you so much for the book recommendation, I'll be sure to check this out.

@GeraldEdgar my apologies for this, where would the correct forum be?

Would $u_n$ and $v_n$ in this solution be sequences of positive reals, thus making $a_n$ also a sequence of positive reals? Using this solution, it seems it could be extended to find a suitable $a_n$ for any number of series we are given.