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#1 This is Cool » A sum for the natural logarithm of any natural number. » 2017-08-18 12:11:26

almighty100
Replies: 1

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,

#2 Re: This is Cool » sums of power sequences » 2017-08-18 10:19:30

Thank you both for the reply.  cmowla replied on his youtube video, as the links he was trying to post for me (to his work on the problem) apparently caused him to be banned... for a only few days, hopefully?

#3 Re: This is Cool » sums of power sequences » 2017-08-16 17:30:18

In 2005 I wanted to actually derive all of the formulas for the sum of integers, squares, cubes, etc.  I arrayed them vertically in a way and found that:





.
.
.

For each of them, the summation on the right hand side was distributed to each term, and after rearranging, collecting like-terms, I was able to derive the formulas for each case.

After finding the sum of the fourth powers, and recognizing the binomial coefficients, I had discovered more generally:

which I have never seen anywhere else before today (which I recognized as I watched your video on youtube for the proof of your theorem).  That video had a link directly to this page, so I signed up and here I am smile

As I continued to find the formulas, I discovered another pattern and derived the following:

Where:

and


is a binomial coefficient.

After discovering this/these formulas, I began to research the sums of powers.  It was then that I learned about Jacob Bernoulli's work on the problem and Bernoulli Numbers.

Once I had learned about them, I found out that I can extract a recursive formula for Bernoulli Numbers out of my formula:

Using this formula, the first Bernoulli number is positive 1/2.  I was proud of myself for discovering these nasty looking formulas.  Although, there are much more efficient methods for arriving with the summation of powers and, of course, Bernoulli Numbers.
The most beautiful that I have seen for the summation of powers is:

or more simply:

where

is the
Bernoulli Number. i.e. the exponents of
act as subscripts.

For example:

This interesting relation of exponents acting as subscripts is referred to as "Umbral Calculus" which, other than this marvelous integral, I have never used before.

At any rate, I'm glad to have found this site.  Today is my first day here.  this is my first (mathematical) post.  I hope you guys are still active.

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