Consider the following sum: $$\huge\sum_{k=1}^{\infty}{ A^{-k} B^{-\frac{\phi^k-1}{\phi-1}} }$$ where $\phi>1$, $A>0$ and $B>1$. The $B$ term dominates the $A$ term for large $k$, so the sum is always finite.
I am looking for a reasonably tight, reasonably simple lower bound on this sum, in terms of $A, B, \phi$.
Obviously one way of obtaining a lower bound is just to truncate the sum, but for small caps, this is unlikely to be tight, and for large caps, it's unlikely to be simple.
[I have given this the power series tag, since this is a power series in $A^{-1}$.]