Newton and Infinite Series

Isaac Newton’s calculus actually began in 1665 with his discovery of the general binomial series (1 + x)ⁿ = 1 + nx + n(n − 1)/2!∙x² + n(n − 1)(n − 2)/3!∙x³ +⋯ for arbitrary rational values of n. With this formula he was able to find infinite series for many algebraic functions (functions y of x that satisfy a polynomial equation p(x, y) = 0). For example, (1 + x)⁻¹ = 1 − x + x² − x³ + x⁴ − x⁵ +⋯ and 1/Square root of√(1 − x²) = (1 + (−x²))^−1/2 = 1 + 1/2∙x² + 1∙3/2∙4∙x⁴+1∙3∙5/2∙4∙6∙x⁶ +⋯.

In turn, this led Newton to infinite series for integrals of algebraic functions. For example, he obtained the logarithm by integrating the powers of x in the series for (1 + x)⁻¹ one by one, log (1 + x) = x − x²/2 + x³/3 − x⁴/4 + x⁵/5 − x⁶/6 +⋯, and the inverse sine series by integrating the series for 1/Square root of√(1 − x²), sin⁻¹(x) = x + 1/2∙x³/3 + 1∙3/2∙4∙x⁵/5 + 1∙3∙5/2∙4∙6∙x⁷/7 +⋯.

Finally, Newton crowned this virtuoso performance by calculating the inverse series for x as a series in powers of y = log (x) and y = sin⁻¹ (x), respectively, finding the exponential series x = 1 + y/1! + y²/2! + y³/3! + y⁴/4! +⋯ and the sine series x = y − y³/3! + y⁵/5! − y⁷/7! +⋯.

Note that the only differentiation and integration Newton needed were for powers of x, and the real work involved algebraic calculation with infinite series. Indeed, Newton saw calculus as the algebraic analogue of arithmetic with infinite decimals, and he wrote in his Tractatus de Methodis Serierum et Fluxionum (1671; “Treatise on the Method of Series and Fluxions”):

I am amazed that it has occurred to no one (if you except N. Mercator and his quadrature of the hyperbola) to fit the doctrine recently established for decimal numbers to variables, especially since the way is then open to more striking consequences. For since this doctrine in species has the same relationship to Algebra that the doctrine of decimal numbers has to common Arithmetic, its operations of Addition, Subtraction, Multiplication, Division and Root extraction may be easily learnt from the latter’s.

For Newton, such computations were the epitome of calculus. They may be found in his De Methodis and the manuscript De Analysi per Aequationes Numero Terminorum Infinitas (1669; “On Analysis by Equations with an Infinite Number of Terms”), which he was stung into writing after his logarithmic series was rediscovered and published by Nicolaus Mercator. Newton never finished the De Methodis, and, despite the enthusiasm of the few whom he allowed to read De Analysi, he withheld it from publication until 1711. This, of course, only hurt him in his priority dispute with Gottfried Wilhelm Leibniz.