Natural log summation representation In working through a problem, I've encountered the need to

Molecca89g

Molecca89g

Answered question

2022-04-24

Natural log summation representation
In working through a problem, I've encountered the need to express
logn=k=1n1log(1+1k)
where logk is the natural logarithm of k. I'm fairly certain the above equality holds (please let me know if otherwise), but I'm not quite sure how to derive it. Any help would be greatly appreciated.
So far, I've tried expressing logn=log(n!)log((n1)!) to no avail.
Many thanks!

Answer & Explanation

Cristal Roth

Cristal Roth

Beginner2022-04-25Added 13 answers

It telescopes after you do a little algebra:
k=1n1log(1+1k)=k=1n1logk+1k
=k=1n1(log(k+1)logk)
=lognlog1
=logn

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