Date of Award

12-2020

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Fejzic, Hajrudin

Abstract

In Calculus we learned that 􏰅Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{􏰅n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at the endpoints a and b. From my recollection, a former instructor informed us to do the value of memorizing these formulas. At the very least, we needed to know that the sum of the first n integers is a polynomial, 1/2 n^2 + · · · of degree 2, the sum of the squares of the first n integers is a polynomial, 1/3 n^3 + · · · of degree 3; lastly, the sum of the cubes of the first n integers is a polynomial, 1/4 n^4 + · · · of degree 4. This resulted in the recognition of a pattern; hence, the generalization; the sum of the r-th powers of the first n integers is a polynomial, 1/(r+1)n^(r+1) + · · · of degree r + 1. Thus, we arrive at the theorem

Theorem 0.0.1. Let r ≥ 0 be an integer. Then there is a unique polynomial, p_r(x) of degree r + 1 with the leading coefficient 1/(r+1) such that for every integer n,

Sum^{n}_{k=1}􏰇k^r =p_r(n).

To some math scholars, these polynomials are called Faulhaber polynomials, named after Faulhaber, a German mathematician, who was one of the first mathematicians to recognize that the sum of r-th powers is indeed a polynomial. His discoveries resulted in “simple” forms of formulating these polynomials when r is odd.

I will prove Theorem 0.0.1, and many other properties of Faulhaber polynomials. For example, I will prove that for all r the coefficient of x^r is 1/2. Thus

p_r(x)= 1/(r+1)x^(r+1)+1/2x^r+···.

The remaining coefficients of this polynomial, do not have a simple form. This is especially the case for the coefficient of x. It turns out that the coefficients of x in p_r(x), are the so-called Bernoulli numbers. I will establish this as well as many related properties about Bernoulli numbers.

I will end my thesis with an interesting observation discussed in [8]. The formula for the sum of cubes is especially interesting giving us the identity

1^{3}+2^{3}+3^{3}+...+(n-2)^{3}+(n-1)^{3}+n^{3} = (1+2+3+...+(n-2)+(n-1)+n)^{2}.

If in both left hand and right-hand sides of this identity we replace the term (n-1) by 2 one would think that the resulting equality

1^{3}+2^{3}+3^{3}+...+(n-2)^{3}+(2)^{3}+n^{3} = (1+2+3+...+(n-2)+(2)+n)^{2}

is false. What was shown in [8] is actually true. Moreover, the pair ((n-2),2) is the only such switch that works for all n.

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