Taylor’s Expansion

Abstract

It was a great triumph in the early years of Calculus when Newton and others discovered that many known functions could be expressed as “polynomials of infinite order” or “power series,” with coefficients formed by elegant transparent laws. The geometrical series for 1/(1 − x) or 1/(1 + x²)

$$\frac{1}{{1\; - \;x}} = 1 + x + {x^2} \cdots + {x^n} + \cdots $$

(1)

$$\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + \cdots + {( - 1)^n}{x^{2n}} + \cdots $$

(1a)

valid for the open interval |x| < 1, are prototypes (see Chapter 1, p. 67).