id: 06604800
dt: j
an: 2016e.00755
au: Winkler, Christopher K.
ti: A method to find the sums of polynomial functions at positive integer
values.
so: Parabola 51, No. 2, 4 p., electronic only (2015).
py: 2015
pu: AMT Publishing, Australian Mathematics Trust, University of Canberra,
Canberra; School of Mathematics \& Statistics, University of New South
Wales, Sydney
la: EN
cc: H20 I20 I30 I50
ut: polynomial functions; polynomial representations; sums of monomials;
positive integers; polynomials of higher degrees; recursion;
simultaneous linear equations; definite integrals
ci:
li: https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-51-2015/issue-2/vol51_no2_1_0.pdf
ab: From the text: When learning the intuition behind definite integration,
calculus students often learn how to find the area under a curve by
using a Riemann sum. Often, when attempting to find the area under
polynomial curves by this method, students are limited by how many
formulas they know for the sums of monomials of a positive integer
degree $n$. The most commonly known formula of this variety is for the
sum of monomials of degree 1, namely $\sum \limits_{i=0}^n
i=\frac{n(n+1)}{2}$. Less common are the formulas for the sum of
monomials of degree 2 and 3, namely $\sum \limits_{i=0}^n
i^2=\frac{n(n+1)(2n+1)}{6}$ and $\sum \limits_{i=0}^n
i^3=\frac{n^2(n+1)^2}{4}$. Most calculus students are taught to
memorize these formulas, and are thus not able to find the sums of
polynomials of degrees higher than 3. Further research into the area
yields Faulhaberâ€™s formula, which involves more complex concepts such
as the Bernoulli numbers, with which students are often unfamiliar. In
this paper, I show a method for deriving these summations for
polynomials of higher degrees without using these complex concepts.
rv: