Infinite product with harmonic numbers

Question

Is it possible to compute an infinite product involving harmonic numbers, such as:

$$\prod\limits_{n=1}^\infty \left(\frac{f^{H_{n}}}{f + f^{H_{n}}}\right)$$

for some constant $f > 1$, where

$$H_{n} = {\sum\limits_{i=1}^n{\frac{1}{i}}}$$?

Answer 1

$H_n = \ln(n) + \gamma + O(1/n)$ as $n \to \infty$, so $f^{H_n} = f^\gamma n^{\ln(f)} (1+O(1/n))$, and $$ 1 - \frac{f^{H_n}}{f + f^{H_n}} = \frac{f}{f + f^{\gamma} n^{\ln(f)} (1 + O(1/n))} \approx f^{1-\gamma} n^{-\ln(f)}$$ Thus the infinite product will converge if $\ln(f) > 1$, i.e. $f > e$.