#### Document Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Mathematics

#### Faculty/School

Faculty of Science

#### First Advisor

Dr. Chester Weatherby

#### Advisor Role

Supervisor

#### Second Advisor

Dr. Shengda Hu

#### Advisor Role

Supervisor

#### Abstract

In this thesis we discuss the various approaches that will be taken to evaluate and find a finite closed form for the sum $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^3+Bn^2+Cn+D)^k}$$ where $B, C, D \in \mathbb{C}$ and $k$ is a positive integer. We begin this thesis by studying the cubic equations and discussing briefly various methods of finding their roots. Cardano's method (1545) for finding the roots of cubic polynomials is explored in detail as this method is used in later parts of the thesis to make calculations while evaluating the sums. Various tools and techniques from Fourier analysis are reviewed for these aid in computing the sums. To obtain finite closed forms for the sums $\sum_{n \in \mathbb{Z}} \frac{1}{(n^3+Bn^2+Cn+D)^k}$, we make use of different methods and approaches from combinatorics and identities involving well-known trigonometric functions.

#### Recommended Citation

Virk, Gagandeep K., "Computing closed forms for the convergent series $\displaystyle\sum_{n \in \mathbb{Z}}\frac{1}{(n^3+Bn^2+Cn+D)^k}$" (2016). *Theses and Dissertations (Comprehensive)*. 1886.

https://scholars.wlu.ca/etd/1886

#### Convocation Year

2016

#### Convocation Season

Fall