Taylor Series Calculator — Expand Functions as Power Series
Compute Taylor series expansions of functions like e^x, sin(x), cos(x) around any center with customizable order.
Taylor Series
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Taylor Polynomial
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Original Function vs Taylor Approximation (center = 0)
Individual Terms
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Number of Terms
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The Formula
The Taylor series expansion represents a function as an infinite sum of terms calculated from its derivatives at a single point. The formula computes each term using the n-th derivative of f evaluated at the center point a, divided by n factorial, multiplied by (x - a)^n. When a = 0, the series is called a Maclaurin series. Taylor series are powerful tools for approximating functions using polynomials, enabling computation of function values, integrals, and solutions to differential equations.
Variable Definitions
Function
The original function being expanded as a Taylor series (e.g., e^x, sin(x), cos(x)).
Center
The point around which the function is expanded. The series approximates the function best near this point.
Order
The highest-power term included in the expansion. Higher order gives better approximation but more terms.
n-th Derivative at a
The n-th derivative of f evaluated at the center point a, determining the coefficient of each term.
n Factorial
The product of all integers from 1 to n, used to normalize each term in the series.
How to Use This Calculator
- 1
Enter a function expression using x as the variable (e.g., e^x, sin(x), cos(x), 1/(1-x)).
- 2
Set the center point a (default 0, which gives the Maclaurin series).
- 3
Choose the order n (default 5) to control how many terms are included.
- 4
Optionally provide an x value to evaluate the Taylor polynomial approximation numerically.
- 5
View the resulting polynomial and each individual term of the expansion.
Quick Reference
| From | To |
|---|---|
| e^x | 1 + x + x²/2! + x³/3! + ... |
| sin(x) | x − x³/3! + x⁵/5! − ... |
| cos(x) | 1 − x²/2! + x⁴/4! − ... |
| 1/(1−x) | 1 + x + x² + x³ + ... (|x| < 1) |
Common Applications
- Approximating transcendental functions like e^x, sin(x), and cos(x) with simple polynomials for computation.
- Solving differential equations by representing unknown functions as Taylor series and matching coefficients.
- Evaluating limits and integrals that are difficult or impossible to compute exactly with elementary methods.
- Modeling physical systems where only a few terms of the expansion capture the dominant behavior near equilibrium.
- Numerical analysis for function approximation and error estimation in scientific computing.
A Taylor series approximates a function as an infinite sum of terms based on its derivatives at a single point
Understanding the Concept
The Taylor series is one of the most important tools in mathematical analysis. It provides a way to represent smooth functions as infinite polynomials, making them easier to study, differentiate, integrate, and evaluate numerically. The fundamental insight is that knowing all the derivatives of a function at a single point is enough to reconstruct the entire function (at least locally). For common functions like e^x, sin(x), and cos(x), the Taylor series have simple, repeating patterns. The approximation improves as more terms are added, and the error term can be bounded using Taylor's theorem. The series converges to the original function within its radius of convergence, which may be finite (like 1/(1-x) which converges only for |x| < 1) or infinite (like e^x and the trig functions which converge everywhere). Taylor series are also the foundation for many numerical methods in science and engineering.
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