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Re: 1 + 2 + ... + n a polynomial how?
Posted:
Apr 1, 2013 12:45 PM


Jussi Piitulainen wrote:
> Is it obvious that 1 + 2 + ... + n is a polynomial of degree 2? How? > > I mean the sum of the first n positive integers. I would like to see > that it is a polynomial of degree 2 _without using_ the fact that it > is equal to n(n + 1)/2. Zeilberger (his new Opinion 129) says Gauss > could have used the polynomiality of the sum to support the equality, > rather than the other way around. > > Thanks for any insight.
I guess Zeilberger was referring to the difference calculus, as used in numerical analysis. If you have a sequence a(n) then Da(n) = a(n)  a(n1). Also if A(n) = sum_{r<=n} then DA(n) = a(n). So in this case the second difference D^A(n) = D(n) = 1 is constant, and it follows that A(n) is a polynomial of degree 2. This goes back long before Gauss, to Newton at least.
 Timothy Murphy email: gayleard /at/ eircom.net tel: +353862336090, +35312842366 smail: School of Mathematics, Trinity College Dublin



