What Is a Taylor Series?

A Taylor series represents a smooth function as an infinite polynomial centered at a point a. The k-th coefficient is the k-th derivative evaluated at a, divided by k!. Truncating the series gives a polynomial that matches the function's value and all its derivatives at the center — and approximates it nearby.

The Series Used Here

eˣ = Σ xᵏ/k!, sin(x) = Σ (−1)ᵏ x^(2k+1)/(2k+1)!, cos(x) = Σ (−1)ᵏ x^(2k)/(2k)!, and ln(1+x) = Σ (−1)^(k+1) xᵏ/k. The first three converge for every real x; the logarithm series converges only for |x| < 1 around a = 0, a classic illustration of a radius of convergence.

Why Taylor Series Matter

Every scientific calculator evaluates sin, cos, and eˣ using exactly this method — polynomial terms that require only multiplication and division. Taylor series also drive physics approximations (the pendulum equation), financial models (the Taylor rule, option Greeks), and numerical analysis, making them one of the most consequential ideas in applied mathematics.