---
title: "Taylor and Maclaurin Series"
description: "Build the power series of a function from its derivatives at a point, use the remainder to say how good a truncation is, and reuse the standard expansions instead of starting over."
canonical: https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO
source: https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO.md
retrieved: 2026-09-12
---

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# Taylor and Maclaurin Series

Build the power series of a function from its derivatives at a point, use the remainder to say how good a truncation is, and reuse the standard expansions instead of starting over.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO

## Ready when they can

- Derive the Maclaurin series of e^x, sin x or 1/(1 - x)
- Obtain a new series by substituting into or differentiating a known one
- Bound the error of a Taylor polynomial with the remainder term

## Lesson: Building functions from their derivatives

A Taylor series rebuilds a smooth function from its derivatives at one center. The coefficient of (x minus a) to the n is the nth derivative at a divided by n factorial. A Maclaurin series is the same idea centered at 0, which keeps the arithmetic simple. For e to the x, every derivative at 0 equals 1, so you get 1 + x + x squared over 2 and on.

**Example.** For sin x the derivatives at 0 cycle through 0, 1, 0, minus 1, so the even terms vanish and degree 3 gives x minus x cubed over 6. For 1 over (1 minus x) every derivative at 0 equals n factorial, so dividing by n factorial leaves coefficient 1 on every power: 1 + x + x squared and on. Stopping after a chosen term gives a Taylor polynomial, like T2(x) = 1 + x + x squared over 2 for e to the x.

Once you know one series you get new ones for free by substitution or term by term moves, valid inside the interval. Replacing x with minus x flips every odd power, turning e to the x into e to the minus x. Multiplying each term by x gives the series for x times e to the x. Differentiating term by term works the same way.

**Tip.** The Lagrange remainder bounds your error with the largest value of the next derivative on the interval, times the power term over (n + 1) factorial. When that bound is tiny, your partial sum is trustworthy. Use it to certify approximations like cos(0.1) before you stop adding terms.

**Recap.** Derivatives at the center give the coefficients, known series give new ones, and the remainder certifies the stop.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Power Series and the Radius of Convergence](https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD)
- [The Binomial Expansion](https://lightmysky.com/learn/mathematics/the-binomial-expansion-mt_z16HZkNZ8c)

## Opens up

- [Molecular Motion Quantised: The Oscillator and the Rotor](https://lightmysky.com/learn/science/molecular-motion-quantised-the-oscillator-and-the-rotor-mt_dHgi-i9pxc)
- [Series Solutions About an Ordinary Point](https://lightmysky.com/learn/mathematics/series-solutions-about-an-ordinary-point-mt_E6NiSd7MED)
- [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8)
- [Polynomial Interpolation and Its Error Term](https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd)
- [First-Order Perturbation Theory](https://lightmysky.com/learn/science/first-order-perturbation-theory-mt_UsbiiQFOAt)