#### 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.

#### Recommended Citation

Agu, Obiamaka L., "Sum of Cubes of the First n Integers" (2020). *Electronic Theses, Projects, and Dissertations*. 1132.

https://scholarworks.lib.csusb.edu/etd/1132

#### Included in

Algebra Commons, Algebraic Geometry Commons, Analysis Commons, Dynamic Systems Commons, Number Theory Commons, Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Other Mathematics Commons