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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 $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

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    Could someone please edit the formula for me so that the integrand of the first integral on the RHS is simply “f”? (I am an advocate of dropping the extra baggage.) – and edit the formula so that the integrand of the second integral is “f’w”? Thanks. (I’m struggling with a Chinese operating system, and a lack of knowledge of whatever markup system the rest of you are using for writing mathematical formulas.)2011-07-04

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It seems the formula should read \sum_{k=0}^nf(k)=\int_0^nf(t)\,dt+{f(0)+f(n)\over2}+\int_0^nf'(t)w(t)\,dt where $w(t)=t-[t]-1/2$. This is how it's given as (1) at the Wikipedia article on the Euler–Maclaurin formula (it's not Euler-Maclaurin, but is a step along the way to the proof of Euler-Maclaurin). This differs from the way Mike has written it in the handling of $n$ and $n+1$ as upper limits, and 0 and 1 as lower limits, but surely those differences are easy to take into account.

I concur with Martin Sleziak that it's called Euler summation (and not Euler-Maclaurin, despite my earlier comment).

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    @Grigory, I have found formulas attributed to Sonin but they don't look like the one we're discussing. Do you have a reference?2011-08-25