Hi Matthew, I just read your email about the behavior of the factorial function and harmonic series for large values of $n$. If you denote by $\gamma \approx 0.5772156649$ the Euler's number, by $e \approx 2.7182818284$ the Euler's constant then you have the well-known Stirling's approximation:$n! = \sqrt{2 \pi n} {\left( \frac{n}{e} \right)}^n \left( 1 + O \left( \frac{1}{n} \right) \right)$where of course I use the classical constant $\pi \approx 3.1415926535$. We also have the following asymptotic expansion:$\sum_{k=1}^n \frac{1}{k} = \ln(n) + \gamma + O \left( \frac{1}{n} \right)$I hope that this answers your question.

Zhen ZHANG

Ph.d Candidates,

School of Computer Science and Technology,

Northwestern Polytechnical University

127 Youyi West Road, 710129, Xi'an, China

Visitor,

School of Computer Science,

The University of Adelaide,

North Terrace, Adelaide, SA5005

Tel: +86-15029259244 +61-4-81383324

Email: zhangzhen@mail.nwpu.edu.cn zhen.zhang@adelaide.edu.au

Residental Address: 160 Marian Road, Glynde, SA5070

Ph.d Candidates,

School of Computer Science and Technology,

Northwestern Polytechnical University

127 Youyi West Road, 710129, Xi'an, China

Visitor,

School of Computer Science,

The University of Adelaide,

North Terrace, Adelaide, SA5005

Tel: +86-15029259244 +61-4-81383324

Email: zhangzhen@mail.nwpu.edu.cn zhen.zhang@adelaide.edu.au

Residental Address: 160 Marian Road, Glynde, SA5070