The calculator will find the Taylor (or power) series expansion of the given function around the given point, with steps shown. You can specify the order of the Taylor polynomial. If you want the Maclaurin polynomial, just set the point to $$$0$$$.

The Taylor and Maclaurin Series Calculator is a tool that expands a function into the Taylor or Maclaurin series. These series are used in calculus to approximate and represent various types of functions as polynomials with an infinite number of terms, making further analysis easier.

## How to Use the Taylor and Maclaurin Series Calculator?

### Input

Enter the function for which you want to calculate the series expansion. Also enter the desired degree (order) of the polynomial and the center point. For the Maclaurin Series, the center point equals $$$0$$$.

### Calculation

Once you've entered the function, order, and center point, click the "Calculate" button.

### Result

The calculator will display the series expansion of the function around the given point up to the specified degree.

## What is the Taylor Series?

In mathematics, the Taylor Series is the expansion of a function into an infinite series. It uses the behavior of the derivatives of a function at a certain point. In other words, the Taylor Series is an approximation of a function by a polynomial function.

**General Form of the Taylor Series:**

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

In this expression:

- $$$f(a)$$$ represents the value of the function at the point $$$x=a$$$.
- $$$f^{\prime}(a)$$$ represents the function's first derivative evaluated at $$$x=a$$$.
- $$$f^{\prime\prime}(a)$$$ represents the function's second derivative evaluated at $$$x=a$$$.
- $$$f^{\prime\prime\prime}(a)$$$ represents the function's second derivative evaluated at $$$x=a$$$.
- The terms continue with higher derivatives as needed, each multiplied by an appropriate power of $$$x-a$$$ and divided by the factorial of the derivative order.

**Key Points about the Taylor Series:**

**Centered at $$$x=a$$$:**The Taylor Series expansion is centered at a specific point $$$x=a$$$. This means that the function's derivatives are evaluated at this point, and the terms are based on how the function behaves around it.**Infinite Series:**The series is infinite, which means it has an unlimited number of terms. The more terms you include, the more accurate the approximation becomes.**Polynomial Approximation:**The Taylor Series approximates a function by a polynomial. The more terms you use, the closer the polynomial will resemble the original function within a small neighborhood of $$$x=a$$$.

For example, let's find the Taylor Series for the natural logarithm function $$$\ln(x)$$$ centered at $$$a=1$$$:

- $$$f(1)=\ln(1)=0$$$
- $$$f^{\prime}(x)=\frac{1}{x}$$$, $$$f^{\prime}(1)=\frac{1}{1}=1$$$
- $$$f^{\prime\prime}(x)=-\frac{1}{x^2}$$$, $$$f^{\prime\prime}(1)=-\frac{1}{1^2}=-1$$$
- $$$f^{\prime\prime\prime}(x)=\frac{2}{x^3}$$$, $$$f^{\prime\prime\prime}(1)=\frac{2}{1^3}=2$$$

Now, we can construct the Taylor Series around $$$x=1$$$:

$$\ln(x)=0+(1)(x-1)+\frac{-1}{2!}(x-1)^2+\frac{2}{3!}(x-1)^3+\ldots$$

Simplify:

$$\ln(x)=(x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3+\ldots$$

There is a pattern in the coefficients of the series, which allows us to write that

$$\ln(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}(x-1)^n$$

This is the Taylor Series expansion of $$$\ln(x)$$$ centered at $$$a=1$$$. It represents the natural logarithm as an infinite sum of the terms involving powers of $$$x - 1$$$.

## What Is the Maclaurin Series?

The Maclaurin Series is a special case of the Taylor Series when $$$a=0$$$.

**General Form of the Maclaurin Series**

The Maclaurin Series for a function $$$f(x)$$$ is represented as follows:

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

In the above expression:

- $$$f(0)$$$ represents the value of the function at $$$x=0$$$.
- $$$f^{\prime}(0)$$$ represents the function's first derivative evaluated at $$$x=0$$$.
- $$$f^{\prime\prime}(0)$$$ represents the function's second derivative evaluated at $$$x=0$$$.
- $$$f^{\prime\prime\prime}(0)$$$ represents the function's third derivative evaluated at $$$x=0$$$.
- The terms continue with higher derivatives as needed, each multiplied by an appropriate power of $$$x$$$ and divided by the factorial of the derivative order.

For example, let's find the Maclaurin Series for the function $$$f(x)=sin(x)$$$:

- $$$f(0)=\sin(0)=0$$$
- $$$f^{\prime}(x)=\cos(x)$$$, $$$f^{\prime}(0)=\cos{0}=1$$$
- $$$f^{\prime\prime}(x)=-\sin(x)$$$, $$$f^{\prime\prime}(0)=-\sin(0)=0$$$
- $$$f^{\prime\prime\prime}(x)=-\cos(0)$$$, $$$f^{\prime\prime\prime}(0)=-\cos(0)=-1$$$

Now, we can write the Maclaurin Series:

$$\sin(x)=0+(1)x+\frac{0}{2!}x^2+\frac{-1}{3!}x^3+\ldots$$

Simplify:

$$\sin(x)=x-\frac{x^3}{6}+\ldots$$

There is a pattern in the coefficients of the series, which allows us to write that

$$\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^{k}}{(2k+1)!}x^{2k+1}$$

This Maclaurin Series expansion represents the sine function as an infinite alternating sum of terms involving the odd powers of $$$x$$$.

## Why Choose Our Taylor and Maclaurin Series Calculator?

### Accuracy

Our calculator provides accurate calculations of the Taylor and Maclaurin Series, ensuring correct results.

### Versatility

Our calculator can handle a wide range of mathematical functions, allowing you to easily approximate functions from various fields.

### Customization

You have the option to specify the order of the series to achieve the desired precision.

### Speed

By automating complex calculations, our calculator saves you time so you can focus on the practical applications of series expansion.

### FAQ

#### What is the difference between the Taylor and Maclaurin Series?

The main difference between the Taylor and Maclaurin Series is the center point. The Taylor Series can be centered at any point $$$x=a$$$, whereas the Maclaurin Series is a special case of the Taylor Series centered at $$$x=0$$$. This simplification makes the Maclaurin Series especially useful when working with functions that are symmetric about the origin.

#### Are there any practical applications for the Taylor and Maclaurin Series?

These series have practical applications in various fields, including physics, engineering, economics, and computer science. They are used for numerical analysis, function approximation, and solving differential equations.

#### What is the Taylor and Maclaurin Series Calculator?

The Taylor and Maclaurin Series Calculator is a tool designed to help users calculate the Taylor or Maclaurin Series expansion of a function.

#### What is the Maclaurin Series?

The Maclaurin Series is a special case of the Taylor Series centered at $$$x=0$$$. In a power series, a function is expressed as the sum of terms involving powers of $$$x$$$, often from $$$x^0$$$ (the constant term) to higher powers.