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# Maclaurin Series Calculator

## Maclaurin Series Lesson

#### Lesson Contents

### Why do we use a Maclaurin Series?

**A Maclaurin series is a special subset of the Taylor series. A Taylor series provides us a polynomial approximation of a function centered on the point a, whereas a Maclaurin series is always centered on a = 0.** Because the behavior of polynomials can be easier to understand than functions such as sin(x), we can use a Maclaurin series to help in solving differential equations, infinite sums, and advanced physics problems. In fact, if we make a Maclaurin series of infinite terms, it will represent the function perfectly.

However, a finite Maclaurin series is just an approximation of the function, where the **accuracy in which the series represents the function is positively correlated with the number of terms in the series.** Carrying out more terms of a Maclaurin series will give us a more accurate representation of the function.

The number of terms in the series is directly linked to the degree of the Maclaurin series. The degree is the maximum *n* value written in the sigma notation of the formula given below. The number of terms in the series is *n* + 1 since the first term is created with *n* = 0. The highest power in the polynomial is *n* = *n*.

### How to Calculate a Maclaurin Series

The formula for calculating a Maclaurin series for a function is given as:

Where *n* is the order, and *f ^{(n)}*(0) is the nth order derivative of f(x) as evaluated at x = 0. The series will be most accurate near the centering point. As we move away from the centering point

*a*= 0, the series becomes less accurate of an approximation of the function.

As we can see, a Maclaurin series may be infinitely long if we choose, but we may also choose to make our series as many or few terms/accurate as we want. We can set a maximum *n* value to make it an *n* order series.

## How the Calculator Works

This calculator is written in the programming language JavaScript (JS) and utilizes a JS-native computer algebra system (CAS). When you click the calculate button, the entire script is run by your device’s internet browser JS engine, allowing for near-instant results.

The CAS employs symbolic computation to create the Maclaurin series expansion. It treats every character as a symbol, rather than a number value. In practice, this avoids computer roundoff error and provides the user with a perfectly accurate analytical solution, being in the form of a mathematical expression.

When the solution is fully calculated, it is converted to LaTeX code. LaTeX is a math markup and rendering language that allows us to graphically display math equations and expressions on a webpage. That final LaTeX solution code is rendered on the page in the answer area.