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Taylor Series Calculator

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

Enter a function:

Variable:

Order:

Enter a point n:

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

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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?

A Taylor series expresses a function as an infinite polynomial built from its derivatives evaluated at a specific point. This representation is extremely useful in calculus, numerical analysis, and applied sciences because it simplifies complex expressions into manageable polynomial terms.

Taylor Series Formula

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

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

n: number of terms included
a: expansion center
f(a): function value at the center
f'(a), f''(a), f'''(a): successive derivatives evaluated at a

Increasing the polynomial degree improves approximation accuracy near the center point.

How to Calculate the Taylor Series?

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?

Function Approximation: Converts difficult functions into simpler polynomials for estimation.
Local Behavior Analysis: Shows how a function behaves near a chosen point.
Scientific Applications: Widely used in physics, engineering, and computer algorithms.
Differential Equation Solutions: Helps approximate solutions when exact methods are complex. Approximation error is measured using the remainder term \(R_n(x)\).

References:

Wikipedia: Taylor Series

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