Hi Matthew, I just read your email about the behavior of the factorial function and harmonic series for large values of 


n

n

. If you denote by 


γ
≈
0.5772156649

\gamma \approx 0.5772156649

 the Euler's number, by 


e
≈
2.7182818284

e \approx 2.7182818284

 the Euler's constant then you have the well-known Stirling's approximation:


n
!
=


2
π
n




(

n
e

)

 
n


(
1
+
O

(

1
n

)

)


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 


π
≈
3.1415926535

\pi \approx 3.1415926535

. We also have the following asymptotic expansion:



∑
 

k
=
1

 
n


1
k

=
ln
(
n
)
+
γ
+
O

(

1
n

)


\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
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