What is the name of the following summation formula?
$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f - \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f'w,$ where $w$ is the “sawtooth” function, defined by $w(x) = (x – (k + 1/2))$, for $k < x <= k + 1$, if $k$ is an integer.
From this formula one can obtain the sum of the first $n$ $k$-th powers. No guessing is necessary, you just turn the crank. However, you have to start at 1 and work your way up. So, if you want the formula for the sum of the first $n$ cubes, say, then you first use this formula to find the formula for the sum of the first $n$ 1-st powers, and then use all this information to find the formula for the sum of the first $n$ squares, and then, finally, use all this information to find the formula for the sum of the first $n$ cubes.
I’ve been calling it Gauss’s Summation Formula, but attributions are often variable, and there might be a more appropriate one that I should be using. I got taken to the woodshed over this. Here is the woodshed link:
Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?