Unit 10: Infinite Sequences & Series

Taylor & Maclaurin Series

Constructing a Taylor or Maclaurin series to approximate a function.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Power Series & Interval of Convergence

Building a Function from Its Own Derivatives

A power series centered at a is just a polynomial with infinitely many terms. Before reading on: if a polynomial P(x) is required to match a function f not just at x = a, but also in its first derivative, second derivative, and every derivative after that at x = a, what would that force each of P's coefficients to be?

Definition — Taylor and Maclaurin Series

Writing P(x) = c₀+c₁(x−a)+c₂(x−a)²+..., and requiring P(a)=f(a), P'(a)=f'(a), P''(a)=f''(a), and so on, forces cₙ = f⁽ⁿ⁾(a)/n! — each repeated differentiation brings down a factorial and eliminates every term but one when evaluated at x = a. The resulting Taylor series is f(x) = Σₙ₌₀^∞ [f⁽ⁿ⁾(a)/n!](x−a)ⁿ; centered at a = 0, it's called a Maclaurin series.

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Worked Example — The Maclaurin Series for eˣ

Since every derivative of f(x) = eˣ is eˣ itself, f⁽ⁿ⁾(0) = 1 for every n. The Maclaurin series is Σxⁿ/n! = 1 + x + x²/2! + x³/3! + ...

Worked Example — Using a Taylor Polynomial to Approximate a Value

Use the first four terms of eˣ's Maclaurin series to approximate e^0.2. P₃(0.2) = 1 + 0.2 + 0.2²/2 + 0.2³/6 = 1 + 0.2 + 0.02 + 0.001333 = 1.221333. The actual value, e^0.2 ≈ 1.221403, differs by only about 0.00007 — the next omitted term, 0.2⁴/4! ≈ 0.0000667, closely matches that leftover error, showing why the very next term is a good rough estimate of how far off a Taylor polynomial's approximation is.

Worked Example — The Maclaurin Series for sin(x)

For f(x) = sin(x): f(0)=0, f'(0)=cos(0)=1, f''(0)=−sin(0)=0, f'''(0)=−cos(0)=−1, f⁗(0)=sin(0)=0 — a pattern that repeats every four derivatives. The series is x − x³/3! + x⁵/5! − x⁷/7! + ... — only odd powers survive, matching sine's odd-function symmetry.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

A Taylor polynomial approximates its function best near the center a, and generally grows less accurate farther away — the same locality every linearization (a Taylor polynomial's simplest, one-term case) already had.

Common Mistakes

Forgetting the n! in the denominator of a Taylor coefficient, using f⁽ⁿ⁾(a) alone instead of f⁽ⁿ⁾(a)/n!.
The factorial comes directly from repeated differentiation of (x−a)ⁿ — omitting it produces coefficients that are systematically too large.

Key Takeaways

A Taylor series' coefficients, f⁽ⁿ⁾(a)/n!, are forced by requiring the polynomial to match every derivative of f at x = a.
A Maclaurin series is a Taylor series centered at 0; eˣ, sin(x), and similar functions have Maclaurin series built directly from their repeating derivative patterns.
A truncated Taylor polynomial approximates its function closely near the center, with the next omitted term roughly indicating the size of the remaining error.

Summary

This closes AP Calculus BC: everything from AP Calculus AB, extended by parametric, vector-valued, and polar calculus, and by infinite sequences and series — connecting a function to a polynomial built entirely from its own derivatives. Together, these representations complete the course's throughline: the same underlying ideas, rates of change and accumulation, expressed through every representation calculus offers.

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