Maclaurin Series — Mathematics Knowledge System

Overview

A Maclaurin series is a Taylor series expanded at the center a = 0. The key expansions -- for e^x, sin x, cos x, 1/(1-x), ln(1+x), and (1+x)^n -- are among the most useful tools in analysis and applied mathematics. Beyond approximation, Maclaurin series provide an elegant alternative to L'Hopital's rule for computing limits of indeterminate forms.

Intuition

A Maclaurin series answers: what polynomial of each degree best approximates f(x) near x=0? The constant term matches f(0), the linear term matches f'(0), the quadratic matches f''(0)/2!, and so on. For functions like e^x and sin x that are infinitely differentiable at 0 with nice derivatives, the series converges everywhere and equals the function exactly.

Formal Definition

Definition

The Maclaurin series of a function f infinitely differentiable at 0 is the power series sum_{n=0}^infty f^(n)(0)/n! * x^n. If this series converges to f(x) in some interval, we say f is represented by its Maclaurin series on that interval.

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\, x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots

Maclaurin series

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots, \quad R = \infty

Exponential function

\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots, \quad R = \infty

Sine function

\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots, \quad R = \infty

Cosine function

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \cdots, \quad R = 1

Geometric series

\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots, \quad R = 1

Natural logarithm

(1+x)^\alpha = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n, \quad R = 1 \text{ (for non-integer } \alpha \text{)}

Binomial series

Notation

Notation	Meaning
$f^{(n)}(0)$	n-th derivative of f evaluated at x = 0
$R$	Radius of convergence of the Maclaurin series
$\binom{\alpha}{n} = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}$	Generalized binomial coefficient

Theorems

Theorem 1: Key Maclaurin Series and Their Radii

e^x = sum x^n/n! (R=infty); sin x = sum (-1)^n x^{2n+1}/(2n+1)! (R=infty); cos x = sum (-1)^n x^{2n}/(2n)! (R=infty); 1/(1-x) = sum x^n (R=1); ln(1+x) = sum (-1)^{n+1} x^n/n (R=1); (1+x)^alpha = sum C(alpha,n) x^n (R=1 for non-integer alpha).

Theorem 2: Euler's Formula via Maclaurin Series

Substituting ix into the Maclaurin series for e^x and separating real and imaginary parts recovers Euler's formula: e^{ix} = cos x + i*sin x.

Theorem 3: Computing Limits via Maclaurin Series

For limits of the form 0/0 or similar indeterminate forms, expanding numerator and denominator as Maclaurin series often gives the limit directly, without differentiation. For example, lim_{x->0} (sin x)/x = lim_{x->0} (x - x^3/6 + ...)/x = 1.

Theorem 4: Uniqueness of Power Series Representation

If a function f has a power series representation sum a_n x^n convergent on (-R, R), then this is the Maclaurin series: a_n = f^(n)(0)/n!. Power series representations are unique.

Worked Examples

1
f(x) = e^x, so f^(n)(x) = e^x for all n. At x=0: f^(n)(0) = 1 for all n.
2
Maclaurin series: sum_{n=0}^infty f^(n)(0)/n! * x^n = sum_{n=0}^infty x^n/n!.
$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
3
By the ratio test, the radius of convergence is R = lim |a_n/a_{n+1}| = lim (n+1)! / n! = infty.

✓ Answer

e^x = 1 + x + x^2/2! + x^3/3! + ... for all x in R.

Practice Problems

Easyapplication

Find the Maclaurin series for sin(x^2) and state its radius of convergence.

Mediumapplication

Use Maclaurin series to compute lim_{x->0} (1 - cos x)/x^2.

Mediumapplication

Find the first 4 nonzero terms of the Maclaurin series for e^x * cos x.

Easyapplication

Using the geometric series 1/(1-x) = sum x^n, find the Maclaurin series for 1/(1+x^2) and hence for arctan x.

Common Mistakes

Common Mistake

Every infinitely differentiable function equals its Maclaurin series.

Not true. The function f(x) = exp(-1/x^2) for x != 0 and f(0) = 0 is infinitely differentiable at 0 with all derivatives equal to 0, so its Maclaurin series is the zero series, which does not equal f(x) for x != 0.

Common Mistake

You can only use Maclaurin series for limits if L'Hopital's rule does not work.

Maclaurin series work for limits regardless of L'Hopital's rule. In fact, they are often faster and more transparent, especially when the limit requires multiple applications of L'Hopital.

Quiz

Historical Background

Brook Maclaurin (1698-1746) popularized the use of Taylor series at a = 0 in his 1742 'Treatise of Fluxions', although the expansion itself was known to James Gregory and Newton earlier. The series named after Maclaurin appeared in Taylor's 1715 work, but it was Maclaurin who systematically applied them. The formula for e^x was essentially known to Euler (1748), and the series for sin and cos appear in Indian mathematics (Madhava, 14th century).

1400s
Madhava of Sangamagrama discovers series for sin and cos (Kerala school)
Madhava
1668
Gregory discovers the series for ln(1+x) and arctangent
James Gregory
1715
Taylor publishes the general Taylor series theorem
Brook Taylor
1742
Maclaurin systematizes the use of Taylor series at a=0 in Treatise of Fluxions
Brook Maclaurin
1748
Euler publishes Introductio, establishing e^x and its series
Leonhard Euler

Summary

The Maclaurin series is the Taylor series at a = 0: f(x) = sum f^(n)(0)/n! * x^n.
Key series: e^x (R=infty), sin x (R=infty), cos x (R=infty), 1/(1-x) (R=1), ln(1+x) (R=1), (1+x)^alpha (R=1).
Euler's formula e^{ix} = cos x + i*sin x follows directly from the Maclaurin series.
Maclaurin series provide an efficient method to compute limits of indeterminate forms by expanding and canceling.

References

BookStewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
BookApostol, T. M. (1967). Calculus, Vol. 1 (2nd ed.). Wiley.
WebsiteWeisstein, E. W. Maclaurin Series. MathWorld.