Taylor Series Calculator

Calculate the Taylor series expansion of any function around a point with step-by-step derivative calculations, interactive comparison graph, and educational explanations.

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Each Taylor term hugs sin x over a wider stretch around x = 0

Enter a function, the expansion point and the order below to get its Taylor polynomial, the derivative steps and a comparison graph.

Taylor Series Expansion Calculator

Quick Examples

f(x) Function to Expand Use: sin, cos, tan, exp, ln, sqrt, **, atan, asin, acos, sinh, cosh

a Expansion Point Use 0 for Maclaurin series

n Order (Degree) Higher = more accurate

x Approximate at (optional) Gives Pₙ(x), the error and its bound

Embed Taylor Series Calculator Widget

About Taylor Series Calculator

Welcome to the Taylor Series Calculator, an advanced mathematical tool that computes the Taylor (or Maclaurin) series expansion of any function around a specified point. This calculator provides step-by-step derivative calculations, a visual comparison graph, and detailed explanations to help you understand polynomial approximations of functions.

What is a Taylor Series?

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Named after the English mathematician Brook Taylor, this powerful technique allows us to approximate complex functions using polynomials, making them easier to analyze, compute, and understand.

The Taylor series provides a bridge between calculus and algebra, transforming transcendental functions like sin(x), e^x, and ln(x) into polynomial expressions that can be evaluated using only addition, subtraction, multiplication, and division.

The Taylor Series Formula

General Taylor Series

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$

Where:

f(x) is the function being approximated
a is the expansion point (center of the series)
f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at point a
n! is the factorial of n (n! = n × (n-1) × ... × 2 × 1)

Maclaurin Series: A Special Case

When the expansion point is zero (a = 0), the Taylor series is called a Maclaurin series. This simplifies the formula since (x - 0)ⁿ = xⁿ:

Maclaurin Series (a = 0)

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$

How to Use This Calculator

Enter your function: Input f(x) using standard mathematical notation. Use ** for exponents, * for multiplication, and function names like sin, cos, exp, ln, sqrt.
Specify the expansion point: Enter the value of a where you want to center the series. Use 0 for a Maclaurin series.
Choose the order: Select how many terms to include (0-20). Higher orders give better approximations but longer polynomials.
Calculate: Click the button to see the Taylor polynomial, step-by-step calculations, and visualization graph.

Common Taylor Series Expansions

Here are frequently used Taylor/Maclaurin series expansions around x = 0:

Function	Maclaurin Series Expansion
$ e^x $	$ 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \cdots $
$ \sin(x) $	$ x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots $
$ \cos(x) $	$ 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots $
$ \ln(1+x) $	$ x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots $
$ \dfrac{1}{1-x} $	$ 1 + x + x^2 + x^3 + x^4 + \cdots $
$ \arctan(x) $	$ x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \cdots $

Understanding Taylor Series Convergence

Not every Taylor series converges for all values of x. The radius of convergence determines the interval where the series accurately represents the function:

e^x: Converges for all real x (infinite radius)
sin(x), cos(x): Converge for all real x (infinite radius)
ln(1+x): Converges for -1 < x ≤ 1
1/(1-x): Converges for |x| < 1

The approximation is most accurate near the expansion point and may diverge as you move farther away, depending on the function's properties.

Applications of Taylor Series

Scientific Computing

Calculators and computers use Taylor series to evaluate transcendental functions. When you press "sin" on your calculator, it likely uses polynomial approximations closely related to Taylor series (or the CORDIC algorithm) to achieve the desired precision.

Physics and Engineering

Taylor series enable linearization of complex systems. For small oscillations, sin(θ) ≈ θ simplifies pendulum equations. In quantum mechanics, perturbation theory uses series expansions to approximate solutions to complex systems.

Numerical Analysis

Taylor series form the foundation of numerical methods for solving differential equations (Euler's method, Runge-Kutta), approximating integrals, and analyzing algorithm complexity.

Signal Processing

Fourier series and transforms, closely related to Taylor series, are essential for analyzing signals, designing filters, and compressing audio/video data.

Frequently Asked Questions

What is a Taylor Series?

A Taylor series is an infinite sum of terms that represents a function as a polynomial. Each term is derived from the function's derivatives evaluated at a single point (the expansion point). The Taylor series allows us to approximate complex functions using simpler polynomial expressions, which is fundamental in calculus, physics, and engineering.

What is a Maclaurin Series?

A Maclaurin series is a special case of the Taylor series where the expansion point is zero (a = 0). It's named after Scottish mathematician Colin Maclaurin. Common Maclaurin series include e^x = 1 + x + x²/2! + x³/3! + ..., sin(x) = x - x³/3! + x⁵/5! - ..., and cos(x) = 1 - x²/2! + x⁴/4! - ...

How do I choose the right order for Taylor series expansion?

The order determines the accuracy of your approximation. Higher orders provide better accuracy but more complex polynomials. For most practical purposes, orders between 3-10 work well. Start with a lower order and increase until the approximation matches your accuracy needs. The error decreases as the order increases, especially near the expansion point.

What functions can be expanded as Taylor series?

Most elementary functions can be expanded as Taylor series, including: polynomial functions, exponential functions (e^x, a^x), logarithmic functions (ln(x), log(x)), trigonometric functions (sin, cos, tan), inverse trigonometric functions (arcsin, arccos, arctan), and hyperbolic functions (sinh, cosh, tanh). The function must be infinitely differentiable at the expansion point.

Why is the Taylor series important in mathematics and science?

Taylor series are essential because they allow us to: approximate complex functions with polynomials for easier computation, solve differential equations, evaluate limits, compute integrals that have no closed form, and implement mathematical functions in calculators and computers. They're fundamental in physics for perturbation theory, signal processing for filter design, and numerical analysis for computational algorithms.

How do I approximate sin(0.2) with a Taylor polynomial?

Use the Maclaurin expansion sin(x) ≈ x − x³/6 + x⁵/120. At x = 0.2 radians, the terms give about 0.198669, close to sin(0.2). Enter sin(x), use 0 as the expansion point, choose an order that includes the x⁵ term, and enter 0.2 in the optional approximation field. Trigonometric Taylor formulas use radians, so convert degrees before using them.

Why can a Taylor polynomial become less accurate farther from its expansion point?

A Taylor polynomial is built from a function’s behavior near its expansion point, so its error often grows as you move farther away. Some functions also have a limited interval where their Taylor series converges to the function. For example, the geometric series for 1/(1−x) converges only when |x| < 1. Increasing the polynomial order does not guarantee a good approximation outside the series’ convergence interval.

Should I use degrees or radians when expanding a trigonometric function?

Use radians, which are the standard input for calculus formulas involving sine, cosine, and tangent. For example, convert 30 degrees to π/6 radians before working with sin(x); entering 30 treats it as 30 radians in standard mathematical notation. If you need a degree-based result, convert the angle first, then evaluate the function or its Taylor polynomial.

Additional Resources

Reference this content, page, or tool as:

"Taylor Series Calculator" at https://MiniWebtool.com/taylor-series-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 19, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

Function	Maclaurin Series Expansion
\( e^x \)	\( 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \cdots \)
\( \sin(x) \)	\( x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots \)
\( \cos(x) \)	\( 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots \)
\( \ln(1+x) \)	\( x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \)
\( \dfrac{1}{1-x} \)	\( 1 + x + x^2 + x^3 + x^4 + \cdots \)
\( \arctan(x) \)	\( x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \cdots \)

Taylor Series Calculator

About Taylor Series Calculator

What is a Taylor Series?

The Taylor Series Formula

Maclaurin Series: A Special Case

How to Use This Calculator

Common Taylor Series Expansions

Understanding Taylor Series Convergence

Applications of Taylor Series

Scientific Computing

Physics and Engineering

Numerical Analysis

Signal Processing

Frequently Asked Questions

Additional Resources

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