Difficulty: Intermediate-Advanced | Prerequisites: Single-variable differentiation and integration (MATH 124/125), familiarity with sequences and series notation

Chapter 16 covers Taylor polynomials and Taylor series, which let you approximate complicated functions with polynomials. This is one of the most tested topics in MATH 126. You need to know how to build Taylor polynomials of any degree, bound the error of the approximation, differentiate and integrate series term by term, and recall the standard Maclaurin series for eˣ, sin x, cos x, 1/(1-x), tan⁻¹x, and -ln(1-x).

TL;DR

Taylor polynomials approximate functions using their derivatives at a single point. The more terms you include, the better the approximation (within the interval of convergence). Error bounds tell you how far off the approximation can be. Six standard Maclaurin series (centred at 0) appear constantly, and you can build new series by differentiating, integrating, or substituting into them.

Key Terms

Taylor polynomial of degree n, Tₙ(x)

A polynomial that matches f and its first n derivatives at a base point b. Tₙ(x) = Σ (from k=0 to n) [f⁽ᵏ⁾(b)/k!] (x - b)ᵏ. Think of it as the best polynomial approximation of f near b.

Maclaurin series

A Taylor series centred at b = 0. The six standard ones (eˣ, sin x, cos x, 1/(1-x), tan⁻¹x, -ln(1-x)) should be memorised.

Error bound (Taylor's inequality)

An upper bound on |f(x) - Tₙ(x)|. If |f⁽ⁿ⁺¹⁾(c)| ≤ M for all c between b and x, then the error is at most [M/(n+1)!] |x - b|ⁿ⁺¹.

Interval of convergence

The set of x-values for which the Taylor series converges to f(x). For example, 1/(1-x) converges for -1 < x < 1.

Core Content: Taylor Polynomials and Error Bounds

First Taylor Polynomial (Linear)

• T₁(x) = f(b) + f'(b)(x - b)

• This is the tangent line approximation

• Linear error bound: if |f''(x)| ≤ M, then |f(x) - T₁(x)| ≤ (M/2)|x - b|²

• The error interval: b - a ≤ x ≤ b + a

Second Taylor Polynomial (Quadratic)

• T₂(x) = f(b) + f'(b)(x - b) + (1/2) f''(b)(x - b)²

• Quadratic error bound: if |f'''(x)| ≤ M, then |f(x) - T₂(x)| ≤ (M/6)|x - b|³

nth Taylor Polynomial (General)

• Tₙ(x) = Σ (from k=0 to n) [f⁽ᵏ⁾(b)/k!] (x - b)ᵏ

• Expanded: f(b) + f'(b)(x-b) + (1/2!)f''(b)(x-b)² + (1/3!)f'''(b)(x-b)³ + ... + (1/n!)f⁽ⁿ⁾(b)(x-b)ⁿ

• Taylor's inequality: if |f⁽ⁿ⁺¹⁾(c)| ≤ M, then |f(x) - Tₙ(x)| ≤ [M/(n+1)!] |x - b|ⁿ⁺¹

Core Content: Operations on Taylor Series

• To differentiate a Taylor series: replace xᵏ with k xᵏ⁻¹ (term-by-term differentiation)

• To integrate a Taylor series: replace xᵏ with xᵏ⁺¹/(k+1) (term-by-term integration)

• You can also substitute, add, and multiply series

Core Content: The Six Standard Maclaurin Series

These are the building blocks. Most exam problems either test these directly or ask you to modify one (by substitution, differentiation, or integration) to produce the series for a related function.

Formulas

Common Misconceptions

• Students often confuse the base point b with a variable. In the formula Tₙ(x) = Σ f⁽ᵏ⁾(b)/k!ᵏ, the derivatives are always evaluated at b (a fixed number), and x is the variable.

• Students sometimes think a Taylor polynomial equals the function everywhere. It does not. The polynomial is only a good approximation near b. How good and how far out depends on the error bound.

• Students forget to find M (the bound on the next derivative) when computing error bounds. M must bound |f⁽ⁿ⁺¹⁾| on the entire interval between b and x, not just at b.

• Students mix up the series for sin x and cos x. Cos starts with 1 (even powers), sin starts with x (odd powers). The signs alternate in both.

Why It Matters / Exam Flags

⚠️ Memorise all six standard Maclaurin series. Exam questions routinely ask you to write them from memory or modify them.

⚠️ Error bound calculations are heavily tested. Practise finding M and applying Taylor's inequality.

⚠️ Term-by-term differentiation and integration of series: know the rules and practise applying them to generate new series from standard ones.

⚠️ The geometric series 1/(1-x) = Σ xᵏ is the simplest and most useful. Many other series can be derived from it by substitution (e.g., replace x with -x² to get 1/(1+x²)).

Quick Self-Test

1. Fill in the blank: T₁(x) = f(b) + f'(b)(________).

   • (x - b).

1. True or false: the Maclaurin series for eˣ converges only for -1 < x < 1.

   • False. It converges for all x.

1. Fill in the blank: sin x = x - x³/3! + x⁵/5! - ________.

   • x⁷/7!.

1. True or false: differentiating a Taylor series term by term gives the Taylor series for f'(x).

   • True (within the interval of convergence).

Practice Q&A

Q: Write the Maclaurin series for e⁻ˣ.

A: Replace x with -x in the series for eˣ: e⁻ˣ = 1 - x + x²/2! - x³/3! + x⁴/4! - ...

Q: Find T₂(x) for f(x) = ln(x) centred at b = 1.

A: f(1) = 0, f'(x) = 1/x so f'(1) = 1, f''(x) = -1/x² so f''(1) = -1. T₂(x) = 0 + 1(x-1) + (-1/2)(x-1)² = (x-1) - (1/2)(x-1)².

Q: Use the error bound to show that |sin(0.1) - T₃(0.1)| < 10⁻⁶, where T₃ is the degree-3 Maclaurin polynomial for sin x.

A: T₃(x) = x - x³/6. The next derivative is f⁽⁴⁾(x) = sin x, so |f⁽⁴⁾| ≤ 1. Error ≤ (1/4!)|0.1|⁴ = (0.0001)/24 ≈ 4.17 × 10⁻⁶. This is about 4 × 10⁻⁶, not quite below 10⁻⁶, so you would need T₅ for that tighter bound. (Good exam practice: check whether the bound actually works.)

Q: Find the Maclaurin series for 1/(1+x²).

A: Start with 1/(1-u) = Σ uᵏ. Let u = -x²: 1/(1+x²) = Σ (-x²)ᵏ = Σ (-1)ᵏ x²ᵏ = 1 - x² + x⁴ - x⁶ + ..., valid for |x| < 1.

Connections to Other Topics

Taylor polynomials are the foundation for numerical methods: they are how calculators compute values of sin, cos, eˣ, and other transcendental functions. The linear Taylor polynomial T₁ is the tangent line from MATH 124, so this chapter generalises an idea you have used since first-year calculus.

The error bounds connect to the mean value theorem and to the rigorous definition of convergence from the series unit in MATH 125. In MATH 126, Taylor series also appear in solving differential equations and in approximating integrals that have no closed-form antiderivative.

Related Terms / Search Tags

Taylor polynomial, Taylor series, Maclaurin series, Taylor's inequality, error bound, remainder term, tangent line approximation, linear approximation, series differentiation, series integration, geometric series, power series, interval of convergence, radius of convergence, eˣ series, sin x series, cos x series, MATH 126, Calculus III, UW