Series of Floor and Ceiling Function—Part I: Partial Summations

Abstract

In this paper, we develop two new theorems relating to the series of floor and ceiling functions. We then use these two theorems to develop more than forty distinct novel results. Furthermore, we provide specific cases for the theorems and corollaries which show that our results constitute a generalisation of the currently available results such as the summation of first n Fibonacci numbers and Pascal’s identity. Finally, we provide three miscellaneous examples to showcase the vast scope of our developed theorems.

Keywords

• ceiling function
• floor function
• Faulhaber’s formula
• Fibonacci numbers
• geometric series
• partial summations