I've been experimenting with the summation of polynomials. My line of attack is to treat the subject the way I would for calculus, but not using limits.
By way of a very simple example, suppose I wish to add the all numbers between 10 and 20 inclusive, and find a polynomial which I can plug the numbers into to get my answer. I suspect its some form of polynomial with degree 2. So I do a integer 'differentiation':
diff(x^2) = x^2-(x-1)^2 = 2x-1
I can see from this that I nearly have my answer, so assuming an inverse 'integration' operation and re-arranging:
(1/2)diff(x^2+int(1)) = x
Now, I know that the 'indefinite integral' of 1 is just x, from 'differentiating' x-(x-1) = 1. So ultimately:
(1/2)(x^2+x) = int(x)
So to get my answer I take the 'definite' integral:
int(x):10,20 = (1/2)(20^2+20)-(1/2)(9^2+9) = 165 (the lower bound needs decreasing by one)
My question is, is there a general way I can 'integrate' any polynomial, in this way?
Please excuse my lack of rigour and odd notation.