On the sum of k-th powers of the first n positive integers

Author

Jancheska Chyles Malilay Malapascua, De La Salle University, ManilaFollow
Arch Raphael Payang Alix, De La Salle University, ManilaFollow

Date of Publication

8-2023

Document Type

Bachelor's Thesis

Degree Name

Bachelor of Science in Mathematics with specialization in Business Applications

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Advisor

Arlene A. Pascasio

Defense Panel Chair

Jose Tristan F. Reyes

Defense Panel Member

Francis Joseph H. Campeña

Abstract/Summary

One of the many functions that has been widely studied and is of interest to mathematicians due to its historic applications is the sum of powers of positive integers. This paper is expository in nature and is based mainly on the article “On the sum of 𝑘^th powers in terms of earlier sums” by Steven J. Miller and Enrique Treviño which appeared in The College Mathematics Journal (2020). For positive integer 𝑛 and nonnegative integer 𝑘, we let 𝑆_𝑘(𝑛) = 1^𝑘 + 2^𝑘 + ⋯ + 𝑛^𝑘. We will give an exposition of the proof of a classical result due to Blaise Pascal which gives the following recursive formula for the sum of powers. Miller and Treviño improved Pascal's formula by proving that for positive integers 𝑛 and 𝑘, the sum of 𝑘^th powers is a polynomial of the sum of 1^st and 2^nd powers. This paper aims to provide a better understanding of sums of powers of positive integers and contribute valuable insights for future work in this area.

Abstract Format

html

Language

English

Format

Electronic

Keywords

Rings of integers

Recommended Citation

Malapascua, J. M., & Alix, A. P. (2023). On the sum of k-th powers of the first n positive integers. Retrieved from https://animorepository.dlsu.edu.ph/etdb_math/31

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Embargo Period

8-15-2023