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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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Mike Jones
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    Isn't this Euler-Maclaurin summation, q.v.? – Gerry Myerson Jul 03 '11 at 04:24
  • @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 03 '11 at 04:30
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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: http://books.google.com/books?id=WZX4GEpxPRgC&pg=PA424&dq=%22Euler+summation+formula%22+sawtooth&hl=en&ei=x_IPTu2FMceD-wbXhP3ZDQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ#v=onepage&q=%22Euler%20summation%20formula%22%20sawtooth&f=false http://books.google.com/books?id=Il64dZELHEIC&pg=PA54&dq=%22Euler+summation+formula%22&hl=en&ei=gfIPTsTNNcOG-wbopbHHDQ&sa=X&oi=book_result&ct=result&resnum=8&ved=0CFgQ6AEwBw#v=onepage&q=%22Euler%20summation%20formula%22&f=false – Martin Sleziak Jul 03 '11 at 04:41
  • 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 03 '11 at 05:51
  • May be it's this: http://en.wikipedia.org/wiki/Abel's_summation_formula –  Jul 03 '11 at 09:23
  • @Chandru, Abel's formula is a bit different. One can deduce Euler's from Abel's, as Apostol does on page 78 of Introduction to Analytic Number Theory. – Gerry Myerson Jul 04 '11 at 00:55
  • 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.) – Mike Jones Jul 04 '11 at 19:29

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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).

Mike Spivey
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Gerry Myerson
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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 04 '11 at 04:39
  • @Mike, so I was right the first time? I can live with that.... – Gerry Myerson Jul 04 '11 at 05:53
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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 04:31