What is the difference between Taylor and Maclaurin series?

A Taylor series expands a function around any point a: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... A Maclaurin series is a special case where a=0: f(x) = f(0) + f'(0)x + f''(0)x²/2! + ...

How do I calculate a Taylor series expansion?

Enter your function, select Taylor series, choose the center point a, and specify the number of terms. The calculator will compute successive derivatives at point a and construct the series: Σ(f⁽ⁿ⁾(a)/n!)(x-a)ⁿ

What is the radius of convergence?

The radius of convergence R is the distance from the center point within which the Taylor series converges to the actual function. For |x-a| R, it diverges.

What functions have known series expansions?

Common functions with known series: e^x = Σ(xⁿ/n!), sin(x) = Σ((-1)ⁿx^(2n+1)/(2n+1)!), cos(x) = Σ((-1)ⁿx^(2n)/(2n)!), ln(1+x) = Σ((-1)^(n+1)xⁿ/n), 1/(1-x) = Σxⁿ for |x|<1

How many terms do I need for a good approximation?

It depends on the function and desired accuracy. Near the center point, fewer terms (5-10) often suffice. For points farther from center or complex functions, more terms (15-20) may be needed. Check the interactive graph to see convergence.

Taylor & Maclaurin Series Calculator

Expand functions into power series with step-by-step solutions

Function f(x) Supported: +, -, *, /, ^, sin, cos, tan, exp, ln, log, sqrt

Series Type

Center Point (a)

Number of Terms (n)

Series Formula

Taylor Series:
f(x) = Σ [f⁽ⁿ⁾(a)/n!] · (x-a)ⁿ

Maclaurin Series:
f(x) = Σ [f⁽ⁿ⁾(0)/n!] · xⁿ

Common Series

e^x = Σ xⁿ/n!
sin(x) = Σ (-1)ⁿx²ⁿ⁺¹/(2n+1)!
cos(x) = Σ (-1)ⁿx²ⁿ/(2n)!
ln(1+x) = Σ (-1)ⁿ⁺¹xⁿ/n
1/(1-x) = Σ xⁿ, |x|<1

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Taylor & Maclaurin Series Calculator

Series Formula

Common Series

Series Expansion

Step-by-Step Solution ▼ Show

Convergence Analysis

Graph Comparison

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Taylor & Maclaurin Series Calculator

Series Formula

Common Series

Series Expansion

Step-by-Step Solution ▼ Show

Convergence Analysis

Graph Comparison

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