Computing closed forms for the convergent series ∑n∈Z1(n3+Bn2+Cn+D)k\displaystyle\sum_{n \in \mathbb{Z}}\frac{1}{(n^3+Bn^2+Cn+D)^k}

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\u27s 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