What Is a Taylor Series?

A Taylor series is a representation of a smooth function as an infinite sum of polynomial terms, each computed from the function's successive derivatives evaluated at a single reference point. Named after the English mathematician Brook Taylor, who published the result in 1715, the series expresses a function f(x) centered at a point a as f(x) = f(a) + f'(a)(x-a) + f''(a)/2!(x-a)² + f'''(a)/3!(x-a)³ + ..., where each term's coefficient is a derivative of f divided by the corresponding factorial. When the expansion point is a = 0, the result is the special case known as a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. Taylor series belong to real and complex analysis and are foundational tools in calculus, numerical methods, signal processing, and control systems engineering.

The utility of the series rests on two properties. First, polynomials are far easier to compute, differentiate, and integrate than most transcendental or nonlinear functions. Second, for many well-behaved functions, truncating the series after a finite number of terms yields a polynomial approximation whose accuracy improves predictably as more terms are included, with the error bounded by Taylor's remainder theorem.

Convergence and Radius of Convergence

A Taylor series does not necessarily converge to the original function for all values of x. The set of values for which the series converges to f(x) is an interval (or disk in the complex plane) centered at the expansion point a, and its half-width is called the radius of convergence. For functions with singularities or branch points, the radius equals the distance from the expansion point to the nearest singularity. Some functions, such as exp(x), sin(x), and cos(x), have infinite radii of convergence and are accurately represented by their Taylor series everywhere. Others, such as 1/(1+x²), converge only on a bounded interval. The NIST Digital Library of Mathematical Functions provides rigorous statements of convergence conditions and coefficient formulas for the standard functions used in applied mathematics and engineering.

Linearization and Small-Signal Analysis

One of the most widely applied uses of the Taylor series in engineering is linearization: replacing a nonlinear function with its first-order Taylor polynomial, f(x) ≈ f(a) + f'(a)(x-a), valid near the expansion point a. In circuit analysis, this approximation models the small-signal behavior of nonlinear devices such as diodes and transistors around a chosen bias point, converting the nonlinear device equations into linear relationships between incremental voltage and current. In control systems, linearization of the nonlinear state equations about an operating point is the standard first step in designing a linear feedback controller using classical or state-space methods. The accuracy of the linear model degrades as the operating deviation from the expansion point increases, so higher-order terms are retained when moderate deviations matter.

Signal Processing and Numerical Methods

Taylor series expansions appear directly in digital signal processing in the construction of digital differentiators and in local frequency analysis. Signal processing using generalized Taylor series expansion extends the classical point-based expansion to expansions about functions, supporting adaptive identification of signals in orthonormal basis frameworks. In numerical methods, finite difference approximations to derivatives are derived from Taylor series by expressing f(x+h) and f(x-h) as expansions and combining terms to isolate the desired derivative at controlled order of error. Taylor series-based numerical methods benchmarking covers the performance of these approaches including generalized finite difference methods for partial differential equations.

Applications

Taylor series have applications in a wide range of fields, including:

• Circuit analysis and analog design, where small-signal models of nonlinear devices are derived from first-order expansions
• Control systems engineering, for linearizing nonlinear plant models around operating points
• Numerical computing, providing the derivation basis for finite difference stencils and integration rules
• Digital signal processing, supporting differentiator design and local frequency estimation
• Physics simulation, approximating potential energy surfaces and propagating solutions to differential equations