Logarithmic Series Evaluation I was trying to generate a direct formula

Vanessa Hensley

Vanessa Hensley

Answered question

2022-01-23

Logarithmic Series Evaluation
I was trying to generate a direct formula for this series but I am not sure whether it is possible to do so.
1ln(1)+2ln(2)+3ln(3)+4ln(4)++(n1)ln(n1)+nln(n)

Answer & Explanation

Kaitlynn Noble

Kaitlynn Noble

Beginner2022-01-24Added 12 answers

By partial summation:
k=1nklogk=(n+12)lognk=1n1(k+12)log(1+1k)
and using the approximation
log(1+1k)=1k12k(k+1)16k(k+1)2+O(1k4) we get:
k=1nklogk=6n2+6n+112lognn24+O(1).
In the O(1)-term hides the logarithm of the Glaisher-Kinkelin constant.

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