Sums of Powers by L'Hopital's Rule
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914901506654208 |
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| author | Dueñez, Eduardo Hamakiotes, Asimina S. Miller, Steven J. |
| author_facet | Dueñez, Eduardo Hamakiotes, Asimina S. Miller, Steven J. |
| contents | For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$; i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's well known that $p_d(n)$ 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 $p_d(n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03624 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sums of Powers by L'Hopital's Rule Dueñez, Eduardo Hamakiotes, Asimina S. Miller, Steven J. Number Theory For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$; i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's well known that $p_d(n)$ 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 $p_d(n)$ 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. |
| title | Sums of Powers by L'Hopital's Rule |
| topic | Number Theory |
| url | https://arxiv.org/abs/2302.03624 |