Maclaurin Polynomial — Definition, Formula & Examples

A Maclaurin polynomial is a polynomial that approximates a function near

x = 0

, built entirely from the function's value and derivatives at zero. Adding more terms usually makes the approximation better, especially for

x

close to 0.

If $f$ is $n$ times differentiable at $0$ , its $n$ th-degree Maclaurin polynomial $P_n$ is the unique polynomial of degree at most $n$ satisfying $P_n^{(k)}(0) = f^{(k)}(0)$ for $k = 0, 1, \dots, n$ . It is the Taylor polynomial of $f$ centered at $a = 0$ .

Key Formula

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(0)}{k!}\,x^k, \qquad |f(x)-P_n(x)|\le \frac{M\,|x|^{n+1}}{(n+1)!}

Where:

$P_n(x)$ = the Maclaurin polynomial of degree n
$f^{(k)}(0)$ = the kth derivative of f evaluated at zero (the 0th derivative is f itself)
$k!$ = k factorial, the product of the whole numbers from 1 to k, with 0! equal to 1
$n$ = the degree, meaning the highest power of x kept
$M$ = any upper bound for the absolute value of the (n+1)th derivative of f on the interval between 0 and x

Worked Example

Problem: Find the 3rd-degree Maclaurin polynomial of

f(x) = e^x

. Use it to estimate

e

, and bound the error.

Step 1: Every derivative of

e^x

e^x

, so

f^{(k)}(0) = e^0 = 1

for all

k

. Substitute into the formula with

n = 3

P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}

Step 2: Since

e = e^1

, evaluate at

x = 1

P_3(1) = 1 + 1 + \frac{1}{2} + \frac{1}{6} = \frac{8}{3} \approx 2.667

Step 3: The 4th derivative is

e^x

, which on

[0, 1]

is at most

e < 3

. Take

M = 3

n = 3

x = 1

|e - P_3(1)| \le \frac{3 \cdot 1^4}{4!} = \frac{3}{24} = \frac{1}{8}

Answer:

P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}

, giving

e \approx \frac{8}{3} \approx 2.667

with error at most

\frac{1}{8} = 0.125

. The actual error is about

0.052

, well within the bound.

Why It Matters

Lagrange error bound questions on Maclaurin polynomials appear regularly on the AP Calculus BC exam and in Calculus II courses. Physicists use the degree-1 polynomial

\sin\theta \approx \theta

to simplify the pendulum equation. Polynomial approximations of this kind are a standard tool in numerical methods for computing functions like

e^x

and

\sin x

Common Mistakes

Mistake: Writing the coefficient of

x^k

f^{(k)}(0)

and forgetting to divide by

k!

. For

e^x

this gives the wrong polynomial

1 + x + x^2 + x^3

Correction: Each coefficient is

\frac{f^{(k)}(0)}{k!}

. The factorial is what makes the

k

th derivative of

P_n

at 0 match

f^{(k)}(0)

, because differentiating

x^k

exactly

k

times produces

k!

Related Terms

Polynomial — the broader concept this term builds on
Derivative of a Power Series — Differentiate known Maclaurin series term by term
Integral of a Power Series — Build new Maclaurin series by integrating known ones
Alternating Series Remainder — Alternative error bound for alternating Maclaurin series
Alternating Series — Maclaurin series of sin and cos alternate
Series Convergence Tests: Ratio, Root, Integral, Comparison — Ratio test finds where Maclaurin series converge
Absolute Convergence — Power series converge absolutely inside their radius