I'm able to derive the formula where $p=1$ and $p=2$, but I don't see how the formula generalizes to higher powers without having to derive each one individually.
Is there an inductive approach? What is the name of this series?
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See https://en.wikipedia.org/wiki/Faulhaber%27s_formula – TheSilverDoe Aug 19 '20 at 16:02
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You can have a look at Faulhaber's formula. – Rémy Sigrist Aug 19 '20 at 16:02
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By 'any $p$', do you mean any $p\in\mathbb{R}$ or $p$ can only be a positive integer? In the former case, I don't anything like Faulhaber's formula exists. – James Moriarty Aug 19 '20 at 16:05
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Another useful resource by Mathologer: Power sum MASTER CLASS. – mwt Aug 19 '20 at 16:06
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1@ShubhrajitBhattacharya Positive integer, I mean. – David Aug 19 '20 at 16:07
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@David, then the Faulhaber formula is the one you are looking for. – James Moriarty Aug 19 '20 at 16:09