Not sure if this is research level? While analyzing a particular algorithm, I came across the following series: $\sum_{k=1}^{m-1}\frac{3^{k}}{\prod_{i=1}^{k-1}2^{2^{i}}}$
Is there a closed form expression that illuminates asymptotic values in terms of $m$ accurately? In particular, I am interested in how slow this function grows compared to $\frac{3^{m}-1}{2}$ (which is the summation if one sets the denominator in each of the sum term to $1$).
Possibly is there an $x$ as a function of that approximates $m$ f(m)$ where $f(m)$ is such that $S_{m} =(\frac{3^{m}-1}{2})^{x}$?(\frac{3^{m}-1}{2})^{f(m)}$?
