Taylor Series Calculator

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):

Taylor Series Calculator

Use this Taylor Series Calculator to represent any function as a Taylor series step by step. You can expand the function by specifying:

The center point (a) around which the series is expanded. By default, this is x = 0.
The degree (n) of the Taylor polynomial, which determines the number of terms considered for approximation.
Error bounds or convergence analysis, which depend on the degree of the polynomial.

Limitation: This calculator is suitable for representing the Taylor series. It cannot analyze convergence or explore alternative series representations.

What Is a Taylor Series?

The Taylor series is an infinite sum of terms derived from a function’s derivatives at a specified point. It is widely used in calculus to approximate the values of complex functions near the chosen point. By representing complex functions as polynomials, computations become simpler and more manageable.

Taylor Series Formula

The general formula for a Taylor series expansion of a function f(x) around a 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: total number of terms included in the expansion
a: center point of expansion
f(a): value of the function at x = a
f'(a), f''(a), f'''(a): first, second, third derivatives, respectively

You can specify the degree n in the calculator to get a finite approximation. A higher degree generally improves the accuracy of the approximation.

How to Calculate the Taylor Series?

Example:

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

Solution:

Taylor series formula:

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

Step 1: Zeroth derivative

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

Step 2: First derivative

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

Step 3: Second derivative

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

Step 4: Construct the polynomial

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

Step 5: Substitute values and simplify

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

Why Use Taylor Series?

Approximating Functions: Partial sums of the series give polynomial approximations of complex functions near a point.
Analyzing Function Behavior: Derivatives in the series reveal how a function behaves near the center point.
Series Representations: Functions like sine, cosine, and exponential functions can be expressed as Taylor series, which is useful in science and engineering.
Solving Differential Equations: Taylor series can approximate solutions of differential equations when exact solutions are difficult. The error of approximation is represented by the remainder function \(R_n(x)\).

References:

Wikipedia: Taylor Series