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Sums of the Powers of Successive Integers

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Abstract

What happens when you sum successive powers of integers? To investigate this, define

$$\displaystyle S_{k,n} = 1 + 2^k + 3^k + \cdots + n^k = \sum _{i=1}^n i^k, \ \ \ \ k=0, 1, \ldots $$

An easy program generates the following table of numeric values for smallk andn.

Not only could nobody but Gauss have produced it,

but it would never have occurred to any but Gauss

that such a formula was possible

Albert Einstein (1879–1955)

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Notes

  1. 1.

    This corresponds to the sum of powers of variety-2 integers, see equation (3.3).

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Authors and Affiliations

  1. (Home address), Beverly, MA, USA

    Randolph Nelson

Authors
  1. Randolph Nelson

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Cite this chapter

Nelson, R. (2020). Sums of the Powers of Successive Integers. In: A Brief Journey in Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-37861-5_7

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