How did Newton use power series in calculus?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Isaac Newton utilized power series as a foundational tool in calculus, viewing the subject as an algebraic extension of arithmetic involving infinite decimals. His approach centered on the discovery of the general binomial series for arbitrary rational exponents, which allowed him to represent complex algebraic functions as infinite series.1
By applying this method, Newton could perform integration and differentiation on these series term-by-term, effectively reducing calculus to algebraic operations. His key applications included:
- Generating Series for Functions: He derived infinite series for functions such as logarithms and inverse sines by integrating the series expansions of their derivatives.1
- Inverse Series: He calculated inverse series to find exponential and sine series, expressing variables in terms of powers of other functions.1
Newton considered these algebraic manipulations of infinite series to be the essence of calculus. He documented these techniques in his manuscripts De Analysi per Aequationes Numero Terminorum Infinitas and Tractatus de Methodis Serierum et Fluxionum, treating the manipulation of variables with infinite terms as analogous to the operations of common arithmetic.1