> I am looking for the origins of efficient calculations of sums
> of the form:
> m m m
> a(k,m) := 1 + 2 + 3 + ... + k
>
> It seems that Gersonides (Rabbi Ben Gershon) knew
> this sums for m=1,2,3.
> Are there earlier instances of the "formulas" for these sums?
I believe Nicomedes has these three, too.
> Do these sums have a standard name nowadays?
Well, the best name is "power sums", but they are much the same
as "Bernoulli polynomials".
Let's define the Bernoulli numbers
1 = B^0, B^1, B^2, B^3, B^4, ...
by the equation " B^n = (B-1)^n " for n =/= 1, where the
quotation marks indicate that the things within them are to be
expaned in "powers of B" before interpreting. Thus we get
B^2 = B^2 - 2B^1 + 1 , whence B^1 = 1/2
B^3 = B^3 - 3B^2 + 3B^1 - 1, whence B^2 = 1/6
then B^3 =0, B^4 = -1/30, B^5 = 0, B^6 = 1/42, B^7 = 0, ... .
Then it's easy to prove (this is "Faulhaber's formula" - see my
"Book of NUmbers, written jointly with Richard Guy) that
1^m + 2^m + ... + x^m = " { (x+B)^(n+1) - B^(n+1) }/(n+1) "
and more generally
x+B
f(1) + f(2) + ... + f(x) = " integral f(t).dt ".
B
John Conway