Find any Integer Power Sum Formula Using Pascal's Triangle (Part 3)

Part 3. THE PROOF. This video concludes this topic of integer power sum formulas with a direct algebraic proof. It is certainly a good exercise for manipulating series, including making substitutions, telescoping sums, and utilizing different "summation algebra identities".

By the end of this video, we will have proved that the "adjusted Pascal's triangle" I have introduced in this video series is indeed all we need to have to construct these power sum formulas from scratch.

I wrote different portions of this proof at different times, but I managed to compile all pieces together to have a 40 line proof.

I first published my work on this proof on March 29, 2013 at the forum linked to below, but my proof there is both incomplete and partially incorrect (minor errors).

[jeremymwilliams]

R = 0 and R = 1 both work with your formula. (even though the indices on the summation start at k = 2, we have 0Ck = 0 =1Ck where nCk is a binomial coefficient).
I discovered a closed form in a similar fashion. actually, i started with something you have derived in your proof. let me explain: I was arraying the sum of powers in a vertical fashion to find:
note that all the summations will be from k = 1 to n
∑k^2 = ∑(2n - 1)(n - k + 1)
the sum of cubes
∑k^3 = ∑(3k^2 - 3k + 1)(n - k + 1)
the sum of fourth powers:
∑k^4 = ∑(4k^2 - 6k^2 + 4k - 1)(n - k + 1)
and after the 5th powers i noticed the pattern of the difference in consecutive ith powers:
∑k^i = ∑(k^i - (k - 1)^i)(n - k + 1)
which is what you have somewhere along in your proof.
after continuing from here for a few more powers, I had discovered another closed form that is also a double sum (i.e. a summation within a summation) that is similar to yours, yet a much different form. actually, from it i was able to extract a recursive formula for Bernoulli Numbers. I did this before I knew what Bernoulli Numbers are. Of course, I didn't realize that I could extract the formula for Bernoulli Numbers until I learned about them. But either way, it is interesting to see your video. that one formula really stood out to me in your proof.

[MrCrisC]

This is pretty Awesome!!!, I have found a similar formula but without the (-1)^k

[jasperchou]

Thank you so much for this video. But I don’t get how do you get the transformation from Pascal triangle in part 1?

[taejusyee]

Hey I just derived a very similar recursive formula for n of the pth power sums but did so geometrically. By making conjectures about higher dimensional shapes, you can arrive at a correct result! Let me know if you are interested in taking a look.

[revilo]

Try to see what happens when you use negative values of n and see if it ends up being equal to alternating power sums