Taylor Series Calculator

- May 09, 2025

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Calculate and visualise Taylor series expansions of mathematical functions. A Taylor series approximates a function using a sum of terms derived from the function's derivatives at a specific point.

Input Function

Function f(x):

Expansion Point (a):

Number of Terms:

Visualisation Range (Min):

Visualisation Range (Max):

Display Options

Decimal Places:

Show exact function

Show approximation error

Show calculation steps

Taylor Series Expansion

Taylor Series

f(x) =

centred at a = 0

Original Function

f(x) =

Remainder Term

Rₙ(x) =

Visual Representation

Term-by-Term Breakdown

n	f⁽ⁿ⁾(a)	Term	Cumulative Sum

Calculation Method

Taylor Series Formula

The Taylor series of a function f(x) centred at x = a is:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + ... + f⁽ⁿ⁾(a)(x-a)ⁿ/n! + ...

Or using summation notation:

f(x) = ∑_n=0^∞ f⁽ⁿ⁾(a)(x-a)ⁿ/n!

When a = 0, this is also called a Maclaurin series.

About Taylor Series

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

Key Concepts

Expansion Point: The point 'a' around which the series is centred
Order of Approximation: The number of terms used in the series
Convergence: How well the series approximates the function as more terms are added
Radius of Convergence: The range of x-values for which the series converges

Common Taylor Series

e^x: 1 + x + x²/2! + x³/3! + ... (at a = 0)
sin(x): x - x³/3! + x⁵/5! - ... (at a = 0)
cos(x): 1 - x²/2! + x⁴/4! - ... (at a = 0)
ln(1+x): x - x²/2 + x³/3 - ... (at a = 0, for |x| < 1)

Applications

Function Approximation: Calculating values of complex functions
Numerical Analysis: Solving differential equations
Physics: Approximating solutions to physical problems
Computer Science: Algorithms for function evaluation
Engineering: Signal processing and control systems

Understanding the Taylor Series

The Taylor series is a powerful mathematical tool used to approximate complex functions. It does this by expressing the function as an infinite sum of terms calculated from its derivatives at a single point. This method allows us to estimate function values and understand their behaviour near a specific point. For example, if we want to evaluate the function at a point close to the expansion point, the Taylor series gives us a good approximation.

Introducing the Taylor series calculator

The Taylor Series Calculator helps you compute and visualize the expansion of various functions easily. By entering a function and selecting parameters, you can see how well the Taylor series approximates the original function. This calculator is user-friendly and designed for students and professionals alike who want to explore mathematical concepts without complex coding or calculations.

Steps to Use the Calculator

Input your function, like sin(x) or e^x.
Select the expansion point, which is where the series will be centred.
Decide on the number of terms to include in the series.
Set the range for visual representation on the graph.

These simple steps let you easily generate a Taylor series and analyze its characteristics.

Features for Enhanced Learning

This calculator comes with several useful features. Users can choose the number of decimal places to display, decide whether to show the exact function, and even view the approximation error. Additionally, you can see the steps taken in calculations. These options not only make the results clearer but also aid in understanding how the Taylor series is formed and its accuracy compared to the original function.

Visualising the Results

The calculator provides a graphical representation of the Taylor series along with the original function. This visualisation helps users see how closely the series matches the function within the specified range. By adjusting the number of terms, you can observe the improvement of the approximation as more terms are added, making it a valuable learning tool for comprehending the convergence of series.

Breaking Down the Calculation

Each term in the series is derived from the derivatives of the function.
The sum of these terms provides a polynomial approximation.
The calculator also displays a term-by-term breakdown, which helps in understanding how each part contributes to the whole.

This breakdown makes it much easier to grasp the process of forming a Taylor series and the significance of each term in the approximation.

Applications of the Taylor Series

Taylor series have numerous applications across different fields. In Mathematics, they are used for approximating functions that are difficult to work with. Engineers utilise these series for signal processing and control systems, while physicists apply them to solve complex problems. The versatility of the Taylor series makes it an essential concept for anyone working with advanced mathematics.

Why Choose This Calculator?

Using the Taylor Series Calculator simplifies the process of exploring and understanding Taylor series. It’s designed to make mathematics accessible and engaging, offering an interactive way to learn about function approximations. Whether you're a student brushing up on Calculus or a professional needing a quick reference, this calculator is an invaluable resource.

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