Closed form for the partial sum \sum_{k=1}^{n}\frac{\ln k}{k}

Krzychau1

Krzychau1

Answered question

2022-01-20

Closed form for the partial sum k=1nlnkk

Answer & Explanation

eskalopit

eskalopit

Beginner2022-01-20Added 31 answers

We have, by partial summation:
k=1nHkk=Hn2k=1n1Hkk+1
hence it follows that:
k=1nHkk=Hk2+Hk(2)2
and since:
Hn=logn+γ+12n+O(1n2)
it follows that:
k=1nlogkk=O(1)+k=1nHkγk=O(1)+12Hk2γHk=12log2n+O(1)
John Koga

John Koga

Beginner2022-01-21Added 33 answers

I am not sure that a closed form exists on the basis of elementary functions.
There is a closed form which involves the gives the Stieltjes constant as well as the generalized Stieltjes constant
k=1nlnkk=γ1γ1(n+1)
what I am not sure you will enjoy.

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