Calculator Online

Home AI Chat AI Math Solver

NEW

Tools ▾ Category ▾

Pricing Sign In

Calculator Online

Your Result is copied!

Taylor Series Calculator

Enter the values to calculate the Taylor series representation of a function.

Enter a function:

Variable:

Enter a point:

Order n:

Calculate the series and determine the error at that point (optional):

Related

Taylor Series Calculator:

Use this Taylor Series Calculator to expand mathematical functions into their Taylor series form with step-by-step results. The tool allows you to approximate complicated functions by converting them into polynomials around a selected point.

Enter the center point (a) where the expansion will occur. If not specified, the calculator uses x = 0.
Select the degree (n) of the Taylor polynomial to control how many terms are included.
View approximation accuracy based on higher-order terms and truncation limits.

Limitation: This calculator generates Taylor polynomial expansions only. It does not test interval convergence or compute alternative power series forms.

What Is a Taylor Series?

The Taylor series is an infinitely long sum of terms derived from the function's derivatives at a specified point.

It is widely used in calculus to approximate the values of complex functions, especially near the chosen point. This Taylor series is particularly useful for representing complex functions with simpler polynomials.

Taylor Series Formula:

The general Taylor series expansion of a function f(x) about the point a is:

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

“n” is the total number of terms included in the Taylor Series
“a” is the center point of the function
𝑓(𝑎) represents the value of the function at the point x = 𝑎
𝑓′(𝑎) is the first derivative
𝑓′′(𝑎) represents the second derivative
𝑓′′′(𝑎) shows the third derivative

The Taylor series is infinite, but you can set the degree of the polynomial (n) to specify. It can also be done with our Taylor series calculator which allows you to specify the “n” value for the approximation (adding a higher degree leads to a more accurate approximation of the function).

How to Calculate the Taylor Series?

To calculate the Taylor series expansion for the function, look at the example using a formula:

Example:

Find the Taylor polynomial of f(x) = √(x² + 4) up to degree n = 2 about x = 1.

Solution:

Using the Taylor polynomial formula:

\( P(x) = \sum_{k=0}^{2} \frac{f^{(k)}(a)}{k!}(x-a)^k \)

Step 1: Function value

\( f(x) = \sqrt{x^2 + 4}, \quad f(1) = \sqrt{5} \)

Step 2: First derivative

\( f'(x) = \frac{x}{\sqrt{x^2 + 4}}, \quad f'(1) = \frac{1}{\sqrt{5}} \)

Step 3: Second derivative

\( f''(x) = \frac{4}{(x^2 + 4)^{3/2}}, \quad f''(1) = \frac{4}{5\sqrt{5}} = \frac{4\sqrt{5}}{25} \)

Step 4: Form the polynomial

\( P(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2}(x-1)^2 \)

Step 5: Substitute values

\( P(x) = \sqrt{5} + \frac{\sqrt{5}}{5}(x-1) + \frac{2\sqrt{5}}{25}(x-1)^2 \)

Why Use Taylor Series?

Approximating Functions: The partial sums of the Taylor series let you approximate functions. These partial sums are (finite) polynomials, and they can be easily computed. With its help, you can create a simple polynomial function that resembles the complex function around a specific point.
Analyzing Function Behavior: The terms of the series provide you with information on how the function behaves near the center point (Each term of the Taylor polynomial is obtained by taking the function's derivatives at a single point)
Series Representations: It helps to represent crucial functions such as sine, cosine, and exponential functions. Therefore it is widely used in science and engineering
Solving Differential Equations: In some cases, the Taylor series can also be used to get the approximate solutions of differential equations. It is particularly beneficial when finding the exact solution looks difficult. The difference between the approximation and actual value of the function f(x) is the remainder. It is denoted by the function Rn(x). Its the error that is associated with approximating a function and can be easily determined by using the Taylor series calculator.

Reference:

Wikipedia: Taylor Series

Related

Calculator Online

Get the ease of calculating anything from the source of calculator online

Links

Home Conversion Calculator About Calculator Online Blog Hire Us Knowledge Base Sitemap Sitemap Two

Support

Email us at