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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:

Proof that $\sum_{k=1}^n k^2$ = $\frac{n(n+1)(2n+1)}{6}$?

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Isn't this Euler-Maclaurin summation, q.v.? –  Gerry Myerson Jul 3 '11 at 4:24
    
@Mike: I tried to edit your question to LaTeX syntax, but I wasn't sure what you meant by "integral(from 1 to n + 1, of f’w)". –  Martin Sleziak Jul 3 '11 at 4:30
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I guess Gerry is right and you meant f'(t)w(t). See the form of Euler summation formulat in thees books: books.google.com/… books.google.com/… –  Martin Sleziak Jul 3 '11 at 4:41
    
If, to find $S_{p} = \sum_{k=1}^{n}k^p$, it requires that we know $S_{1}, \ldots, S_{p-1} $, then it isn't that effective, surely? I can think of an extremely simple way to do that -- or, at least, when p is odd! –  Lyrebird Jul 3 '11 at 5:51
    
May be it's this: en.wikipedia.org/wiki/Abel's_summation_formula –  user9413 Jul 3 '11 at 9:23
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1 Answer 1

up vote 7 down vote accepted

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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I would say it's the Euler-Maclaurin summation formula with a first-order remainder term - although it is also a step on the way to proving the infinite series version of the formula. (An analogy is Taylor's formula: You can express it with a remainder term or as an infinite series.) And I've seen it called both "Euler's summation formula" and "Euler-Maclaurin summation formula." In fact, once I called it the former in a paper and was told by a referee to change it to the latter. –  Mike Spivey Jul 4 '11 at 4:39
    
@Mike, so I was right the first time? I can live with that.... –  Gerry Myerson Jul 4 '11 at 5:53
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(Essentially it's Euler-Maclaurin indeed but) I've also heard name “Sonin(e) formula” for this particular variant. –  Grigory M Aug 24 '11 at 21:04
    
@Grigory, I have found formulas attributed to Sonin but they don't look like the one we're discussing. Do you have a reference? –  Gerry Myerson Aug 25 '11 at 4:31
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