Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

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

share|improve this question

1 Answer 1

up vote 3 down vote accepted

$H_n = \ln(n) + \gamma + O(1/n)$ 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. $f > e$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.