On the sum of k-th powers of the first n positive integers - Malapascua, Jancheska Chyles Malilay (Animo Repository, 2023)
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 𝑘th 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 𝑆𝑘(𝑛) = 1𝑘 + 2𝑘 + ⋯ + 𝑛𝑘. 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 𝑘th powers is a polynomial of the sum of 1st and 2nd 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.