Taylor & Maclaurin Series
A Taylor series is a way of expressing any smooth function as an infinite polynomial — a sum of terms involving powers of (x − a), where a is a chosen center point. The key insight is that if we know all the derivatives of a function at a single point, we can reconstruct the entire function as a power series. This bridges the gap between complicated functions like sin x, eˣ, and ln(1+x) and the simple arithmetic of polynomials. A Maclaurin series is simply a Taylor series centered at a = 0. Because zero is the most convenient center for many common functions, Maclaurin series appear constantly in calculus, physics, and engineering. For example, the Maclaurin series for eˣ, sin x, and cos x are among the most important formulas in all of mathematics, and they reveal a deep connection between exponential and trigonometric functions. Taylor and Maclaurin series are used both for exact representation (when the infinite series converges to the function) and for practical approximation (when we truncate the series to a finite polynomial). Understanding how quickly these partial sums converge, and how large the error can be, is central to applying these series in real problems.
- The Core Idea: Matching Derivatives
- The General Formula
- Key Maclaurin Series to Memorize
- Radius and Interval of Convergence
- Taylor Polynomials and Error
The Core Idea: Matching Derivatives
Suppose we want a polynomial P(x) = c₀ + c₁(x−a) + c₂(x−a)² + c₃(x−a)³ + ⋯ that behaves exactly like f(x) near x = a. If P(a) = f(a), then c₀ = f(a). If P′(a) = f′(a), then c₁ = f′(a). If P″(a) = f″(a), then 2c₂ = f″(a), so c₂ = f″(a)/2. In general, differentiating n times and evaluating at a gives cₙ = f⁽ⁿ⁾(a)/n!. This matching of all derivatives at one point is the entire foundation of Taylor series.
The General Formula
The Taylor series of f(x) centered at x = a is: f(x) = Σₙ₌₀^∞ f⁽ⁿ⁾(a)/n!ⁿ = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + f‴(a)(x−a)³/3! + ⋯. When a = 0, this becomes the Maclaurin series: f(x) = Σₙ₌₀^∞ [f⁽ⁿ⁾(0)/n!]xⁿ.
Key Maclaurin Series to Memorize
Three Maclaurin series are essential: (1) eˣ = 1 + x + x²/2! + x³/3! + ⋯ = Σₙ₌₀^∞ xⁿ/n!, valid for all x. (2) sin x = x − x³/3! + x⁵/5! − ⋯ = Σₙ₌₀^∞ (−1)ⁿ x^(2n+1)/(2n+1)!, valid for all x. (3) cos x = 1 − x²/2! + x⁴/4! − ⋯ = Σₙ₌₀^∞ (−1)ⁿ x^(2n)/(2n)!, valid for all x. These three series together yield Euler's famous identity e^(iπ) + 1 = 0 when complex numbers are allowed.
Radius and Interval of Convergence
A Taylor series only equals f(x) where it converges. The set of x-values for which the series converges is the interval of convergence, with half-width R called the radius of convergence. For eˣ, sin x, and cos x, R = ∞ (they converge everywhere). For ln(1+x) = x − x²/2 + x³/3 − ⋯, the radius is R = 1, so the series converges only for −1 < x ≤ 1. The ratio test is the standard tool for finding R: if lim|aₙ₊₁/aₙ| = L then R = 1/L.
Taylor Polynomials and Error
The nth-degree Taylor polynomial Tₙ(x) is the partial sum up to the xⁿ term. It approximates f(x) near a. The error (remainder) Rₙ(x) = f(x) − Tₙ(x) satisfies the Lagrange remainder formula: |Rₙ(x)| ≤ M|x−a|^(n+1)/(n+1)!, where M is the maximum of |f⁽ⁿ⁺¹⁾| between a and x. This gives a concrete bound on how accurate your polynomial approximation is.
The Key Equation
- f^{(n)}(a) — nth derivative at a (depends on f): The nth derivative of the function f evaluated at the center point a; it determines the coefficient of the nth term.
- n! — n factorial (dimensionless): The product 1 × 2 × 3 × ⋯ × n; appears because repeated differentiation of xⁿ produces n! as a constant factor.
- a — center of expansion (same as x): The point around which the series is built. Setting a = 0 gives the Maclaurin series.
- (x-a)^n — power term (same as x to the nth power): The polynomial basis; each term measures how far x is from the center, raised to the nth power.
Worked Example
Problem. Find the Maclaurin series for f(x) = eˣ and use the degree-3 Taylor polynomial T₃(x) to approximate e^0.5. Estimate the error using the Lagrange remainder.
Answer. T₃(0.5) ≈ 1.6458, with error less than 0.0052
Origins · 1 minBrook Taylor and how this was discovered
Brook Taylor published his method of expanding functions as power series in his 1715 work Methodus Incrementorum Directa et Inversa. The theorem bearing his name states that any sufficiently smooth function can be written as an infinite series of polynomial terms whose coefficients are determined by the function's derivatives at a single point. Although the concept was also known to Johann Bernoulli and James Gregory, Taylor's systematic formulation made it a cornerstone of calculus. Taylor was also an accomplished musician and painter, and he published a major work on the mathematical theory of perspective in art — he used calculus to study how the eye perceives depth and vanishing points.
Related Concepts
- Introduction to Power Series — Taylor series are a special type of power series; understanding the general form Σcₙ(x−a)ⁿ, radius of convergence, and term-by-term differentiation is necessary before specialising to Taylor series.
- Higher-Order Derivatives — Every coefficient in a Taylor series is a higher-order derivative divided by a factorial; fluency with computing f″, f‴, and f⁽ⁿ⁾ is essential.
- Convergence Tests for Series — The ratio test and comparison test are used to find the radius of convergence of a Taylor series and determine where it equals the original function.
- Geometric Series — The geometric series formula 1/(1−x) = Σxⁿ is itself a Maclaurin series and serves as a template for deriving other series through substitution and integration.