2023/02/07 by Eduardo Duéñez, Dueñez, Eduardo, Asimina S. Hamakiotes +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2302.03624
openalex publication_date 2023/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a positive integer d, let pd(n) := 0d + 1d + 2d + ⋯ + nd; i.e., pd(n) is the sum of the first dth-powers up to n. It's well known that pd(n) is a polynomial of degree d+1 in n. While this is usually proved by induction, once d is not small it's a challenge as one needs to know the polynomial for the inductive step. We show how this difficulty can be bypassed by giving a simple proof that pd(n) is a polynomial of degree d+1 in n by using L'Hopital's rule, and show how we can then determine the coefficients by Cramer's rule. This illustrates a general principle and the point of our paper: there's more than one path to a goal, different approaches have their advantages and disadvantages, and the more techniques one knows, the more likely one can successfully attack a problem.