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Sums of Powers of Positive Integers

Author(s): 
Janet Beery (University of Redlands)

Students often encounter formulas for sums of powers of the first n positive integers as examples of statements that can be proved using the Principle of Mathematical Induction and, perhaps less often nowadays, in Riemann sums during an introduction to definite integration. In either situation, they usually see only the first three such sum formulas,

$$1 + 2 + 3 + \cdots + n = {{n(n + 1)}\over 2}$$

$$1^2 + 2^2 + 3^2 + \cdots + n^2 = {{n(n + 1)(2n + 1)} \over 6}$$

and

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = {{n^2 (n + 1)^2 } \over 4}$$

for any positive integer n.

Formulas for sums of integer powers were first given in generalizable form in the West by Thomas Harriot (c. 1560-1621) of England. At about the same time, Johann Faulhaber (1580-1635) of Germany gave formulas for these sums up to the 17th power, far higher than anyone before him, but he did not make clear how to generalize them. Pierre de Fermat (1601-1665) often is credited with the discovery of formulas for sums of integer powers, but his fellow French mathematician Blaise Pascal (1623-1662) gave the formulas much more explicitly. The Swiss mathematician Jakob Bernoulli (1654-1705) is perhaps best and most deservedly known for presenting formulas for sums of integer powers to the European mathematical community. His was the most useful and generalizable formulation to date because he gave by far the most explicit and succinct instructions for finding the coefficients of the formulas.

In this article, we first recount some of the early history of formulas for sums of integer powers. We then explore how each of the mathematicians listed above developed, understood, and presented their formulas. Along the way, we shall see that the phrase “general formula” is relative terminology: during the late sixteenth and early seventeenth centuries mathematicians were just beginning to replace verbal descriptions by symbolic representations and we may be surprised to discover that some mathematicians’ “formulas” were given entirely in words and/or were given just for the first few exponents with the remaining cases covered by “et cetera”. At the end of each section, we present exercises and activities designed to help students develop a deeper understanding of the ideas and methods of each of the mathematicians we discuss.

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