Taylor Series

You expand the function around a point a and sum terms of the form f^(n)(a)/n! · (x−a)^n over all n from zero to infinity. Each coefficient contains the n-th derivative at the expansion point, divided by the factorial of n. In practice you first compute the derivatives at a, divide by n! and multiply by the power (x−a)^n. Example sine around a = 0: sin x = x − x³/6 + x⁵/120 − …, because the derivatives alternate between sine and cosine. For x = 0.1 even the first approximation gives sin(0.1) ≈ 0.09983, very close to the exact value.

If you truncate the Taylor series after the term for n = 1, you are left with f(a) + f′(a)·(x−a). This is exactly the equation of the tangent to the function at the point a, that is the linear approximation or linearization. The constant term f(a) fixes the function value, the linear term f′(a)·(x−a) the slope. Near the expansion point this line describes the function very well, because higher powers of (x−a) are still very small there. Physics and engineering use this approximation constantly, for example for small deflections of a pendulum, where sin x is replaced by x. Higher Taylor terms improve the accuracy further.

A Taylor series describes a function only locally and does not automatically agree with it everywhere. Two conditions must be met: the series must converge, that is stay within its radius of convergence, and the remainder must tend to zero as the number of terms grows. Outside the radius of convergence the series diverges and gives useless values. There are even functions whose Taylor series converges but not to the function itself. A common mistake is to assume that finitely many terms are exact everywhere. For practical approximations you therefore choose an expansion point near the point of interest and estimate the error via the remainder.

Taylor series replace complicated functions with polynomials that are much easier to compute with. Calculators and computers evaluate sine, cosine, exponential and logarithm functions internally via such series, because only addition and multiplication are needed. In physics one linearizes difficult relationships with the first term, for example the pendulum equation for small angles. In error analysis Taylor is used to estimate how measurement uncertainties propagate. Limits of indeterminate form can also be found through series expansion. Finally Taylor is used in numerical methods to approximate derivatives and integrals. The idea of locally replacing a function with a polynomial is one of the most powerful tools of calculus.

A Maclaurin series is simply the special case of a Taylor series with the expansion point a = 0. While the general Taylor series expands around an arbitrary point a and contains powers of (x−a), the Maclaurin series expands around the origin and contains pure powers of x. This makes the formulas simpler, because the derivatives only need to be evaluated at zero. The well-known series for e^x, sin x and cos x are all Maclaurin series. You choose the expansion point so that it lies near the point of interest and the derivatives are easy to compute there; for many standard functions this is exactly the origin.