Sums of Powers of Positive Integers - Conclusion

Sums of powers of positive integers have been of interest to mathematicians since antiquity. Over the years, mathematicians in various places have given verbal formulas for the sum of the first n positive integers, the sum of the squares of the first n positive integers, the sum of the cubes of the first n positive integers, and so on. Beginning as early as the tenth or eleventh century, general methods existed. However, since each sum depended on the sums of the lower powers and required extensive new calculation, often done entirely verbally, in practice, these general methods did not result in calculation of formulas for sums of very high powers. Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber used some symbolic notation and must have spent months, if not years, calculating formulas for sums of powers up to the 17^th power for his 1631 Academia Algebrae. Jakob Bernoulli also may have spent months or years calculating formulas for sums of powers up to the tenth power, but at some point he hit upon the pattern needed to compute relatively quickly and easily the coefficients of the formula for the sum of the cth powers for any positive integer c. Although his method required one to have computed sums of lower powers, or at least to have recorded the Bernoulli numbers, it was efficient enough that Bernoulli himself accomplished the following amazing feat.

With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500. (Smith, p. 90)

Today we might write Bernoulli’s formula for the sum of the cth powers as $$\sum_{k = 1}^n {k^c } = {1 \over {c + 1}}\sum_{m = 0}^c {c+1\choose m}B_m n^{c + 1 - m} $$

with the Bernoulli numbers B_m defined recursively by $$B_0 = 1;\, B_1 = {1 \over 2};\, {1 \over {m + 1}}\sum_{i = 0}^m {m+1\choose i}B_i = 1$$

for m even and at least 2; and B_m = 0 for m odd and at least 3. Actually, $${1 \over {m + 1}}\sum_{i = 0}^m {m+1\choose i}B_i = 1$$

defines the Bernoulli numbers recursively for every nonnegative integer m. The first several Bernoulli numbers defined by this formula are B₀ = 1, B₁ = 1/2, B₂ = 1/6, B₃ = 0, B₄ = –1/30, B₅ = 0, B₆ = 1/42, B₇ = 0, B₈ = –1/30, B₉ = 0, B₁₀ = 5/66, B₁₁ = 0, …. For more modern formulas for sums of powers and Bernoulli numbers, see the article by Apostol, especially pp. 178-179.

Acknowledgments: We are grateful to Patricia Cornez (instructor) and Michael Camp (student), of the University of Redlands Department of Computer Science, for preparing the animations for this article; to John Navarrette, student of mathematics and of German at the University of Redlands, for his assistance with translation; and to Sandra Richey, of the University of Redlands Library, for obtaining books and journals from libraries near and far. We thank the British Library and Huntington Library for the use of manuscripts and rare books, the University of Dresden Digital Library for the use of a digital copy of Academia Algebrae, and Columbia University for permission to use a digital image from Academia Algebrae. Finally, we thank the editors of Convergence, Victor Katz and Frank Swetz, for their encouragement and assistance.