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# ClosedformforGrowthofaparticular sequence

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}$?(\frac{3^{m}-1}{2})^{f(m)}$?

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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 $m$ such that $S_{m} =(\frac{3^{m}-1}{2})^{x}$?

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