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#1 2017-08-18 12:11:26

almighty100
Member
Registered: 2017-08-16
Posts: 5

A sum for the natural logarithm of any natural number.

Consider the goemetric sequence:


which is valid for all

Taking the integral of both sides with respect to r, we have:

and if we let


note

 
for all natural numbers
  (so the restriction on
is satisfied)

Therefore:

or more compactly:

for any natural number

So, for example,

Last edited by almighty100 (2017-08-18 12:12:12)

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#2 2017-08-18 23:40:21

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,123
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Re: A sum for the natural logarithm of any natural number.

Thanks for the contribution! You will have a constant of integration lingering around, but this is easily dealt with by using the initial condition on your geometric series (or any suitable value of
).

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