Math Is Fun Forum

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

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

Where:

and

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

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.