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Maclaurin Series Calculator

Expand functions in Maclaurin series (Taylor at x=0)

Select Function

e^x sin(x) cos(x) 1/(1-x)

Terms (0 to n): n =

Maclaurin Series Formula

f(x) = sum_{n=0}^{inf} f^(n)(0) * x^n / n!

Term 0: f(0) (constant)

Term 1: f(0)*x (linear)

Term n: f^(n)(0)*x^n/n!

The Maclaurin series is the most commonly used power series representation. It approximates functions near zero using derivative information at the origin. Many functions have well-known Maclaurin series used in calculus and numerical computing.

⚠ The series converges to f(x) only within its interval of convergence. Check the radius for each function type.

What Is a Maclaurin Series?

A Maclaurin series represents a function as an infinite polynomial centered at 0. It is the simplest form of Taylor series. Calculating it requires only derivatives at 0, making it easier than general Taylor series for functions well-behaved near the origin.

Definition

f(x)=sum f^(n)(0)*x^n/n!. Taylor at 0. Approximates function behavior near the origin.

e^x Series

e^x=1+x+x^2/2!+x^3/3!+... Converges for all x. All derivatives at 0 are 1.

sin/cos Series

sin(x)=x-x^3/3!+x^5/5!-..., cos(x)=1-x^2/2!+x^4/4!-... Only odd/even powers. Converge everywhere.

Interval of Convergence

e^x, sin, cos: (-inf,inf). 1/(1-x): (-1,1). ln(1+x): (-1,1]. Geometry series converges only within radius 1.

Teaching Example: e^x at n=5. Terms: 1 (n=0), x (n=1), x^2/2 (n=2), x^3/6 (n=3), x^4/24 (n=4), x^5/120 (n=5). Series: 1+x+x^2/2+x^3/6+x^4/24+x^5/120.

Applications

Calculators Physics Engineering Limits Approximations

Frequently Asked Questions

What is Maclaurin series?▼

Taylor series at x=0. f(x)=sum f^(n)(0)*x^n/n!. Simplest power series representation.

e^x Maclaurin series?▼

e^x=1+x+x^2/2!+x^3/3!+...+x^n/n!+... All derivatives are 1 at 0. Converges for all x.

Sin Maclaurin series?▼

sin(x)=x-x^3/3!+x^5/5!-...+(-1)^n x^(2n+1)/(2n+1)!+... Only odd powers with alternating signs.

Interval of convergence?▼

e^x, sin, cos: all reals. 1/(1-x): |x|<1. ln(1+x): (-1,1]. Depends on function singularities.

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