Maclaurin Series Calculator is a free online tool that displays the expansion series for the given function. STUDYQUERIES’s online Maclaurin series calculator tool makes the calculation faster, and it displays the expanded series in a fraction of seconds.

**How to Use the Maclaurin Series Calculator?**

The procedure to use the Maclaurin series calculator is as follows:

**Step 1:**Enter two functions in the respective input field**Step 2:**Now click the button “Calculate” to get the result**Step 3:**Finally, the expansion series for the given function will be displayed in the new window

Maclaurin Series Calculator

**How does 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.

**What is Maclaurin Series?**

In mathematics, the Maclaurin series is defined as the extended series of specific functions. In this series, the approximated value of the given function can be determined as the sum of the derivatives of any function. When the function expands to zero instead of other values a = 0.

**Maclaurin Series Formula:**

The formula used by the Maclaurin series calculator for computing a series expansion for any function is:

$$ Σ^∞_{n=0} \frac{f^n (0)} {n!} x^n $$

Where f^n(0) is the nth order derivative of function f(x) as evaluated and n is the order x = 0. The series will be more precise near the center point. As we shift from the center point a = 0, the series becomes less precise of an approximation of the function.

However, an online Arithmetic Sequence Calculator helps you to calculate the Arithmetic sequence, nth value, and some of the arithmetic sequence.

**Overview of Taylor/Maclaurin Series**

Consider a function \(f\) that has a power series representation at \(x=a\). Then the series has the form

$$\sum_{n=0}^∞c_n(x−a)^n=c_0+c_1(x−a)+c_2(x−a)^2+ \dots \longrightarrow(eq.\ 1)$$

What should the coefficients be? For now, we ignore issues of convergence but instead focus on what the series should be if one exists. We return to discuss convergence later in this section. If the series Equation (1) is a representation for \(f\) at \(x=a\), we certainly want the series to equal \(f(a)\) at x=a . Evaluating the series at \(x=a\), we see that

$$\sum_{n=0}^∞c_n(x−a)^n=c_0+c_1(a−a)+c_2(a−a)^2+\dots=c_0.\longrightarrow(eq.\ 2)$$

Thus, the series equals \(f(a)\) if the coefficient \(c_0=f(a)\). In addition, we would like the first derivative of the power series to equal \(f'(a)\) at \(x=a\). Differentiating Equation (2) term-by-term, we see that

$$\frac{d}{dx}\left(\sum_{n=0}^∞c_n(x−a)^n\right)=c_1+2c_2(x−a)+3c_3(x−a)^2+\dots.\longrightarrow(eq.\ 3)$$

Therefore, at \(x=a\), the derivative is

$$\frac{d}{dx}\left(\sum_{n=0}^{∞}c_n(x−a)^n\right)=c_1+2c_2(a−a)+3c_3(a−a)^2+\dots=c_1.\longrightarrow(eq.\ 4)$$

Therefore, the derivative of the series equals \(f'(a)\) if the coefficient \(c_1=f'(a)\). Continuing in this way, we look for coefficients cn such that all the derivatives of the power series Equation (4) will agree with all the corresponding derivatives of \(f\) at \(x=a\). The second and third derivatives of Equation (3) are given by

$$\frac{d^2}{dx^2} \left(\sum_{n=0}^∞c_n(x−a)^n \right)=2c_2+3⋅2c_3(x−a)+4⋅3c_4(x−a)^2+\dots\longrightarrow(eq.\ 5)$$

and

$$\frac{d^3}{dx^3} \left( \sum_{n=0}^∞c_n(x−a)^n \right)=3⋅2c_3+4⋅3⋅2c_4(x−a)+5⋅4⋅3c_5(x−a)^2+⋯\longrightarrow(eq.\ 6)$$

Therefore, at \(x=a\), the second and third derivatives

$$\frac{d^2}{dx^2}\left(\sum_{n=0}^∞c_n(x−a)^n\right)=2c_2+3⋅2c_3(a−a)+4⋅3c_4(a−a)^2+\dots=2c_2\longrightarrow(eq.\ 7)$$

and

$$\frac{d^3}{dx^3} \left(\sum_{n=0}^∞c_n(x−a)^n\right)=3⋅2c_3+4⋅3⋅2c_4(a−a)+5⋅4⋅3c_5(a−a)^2+\dots =3⋅2c_3\longrightarrow(eq.\ 8)$$

equal \(f′′(a)\) and \(f′′′(a)\) , respectively, if \(c_2=\frac{f′′(a)}{2}\) and \(c_3=\frac{f′′′(a)}{3⋅2}\). More generally, we see that if \(f\) has a power series representation at \(x=a\), then the coefficients should be given by \(c_n=\frac{f^{(n)}(a)}{n!}\). That is, the series should be

$$\sum_{n=0}^∞\frac{f^{(n)}(a)}{n!}(x−a)^n=f(a)+f′(a)(x−a)+\frac{f”(a)}{2!}(x−a)^2+\frac{f”'(a)}{3!}(x−a)^3+⋯\\longrightarrow(eq.\ 9)$$

This power series for \(f\) is known as the Taylor series for \(f\) at \(a\). If \(x=0\), then this series is known as the Maclaurin series for \(f\).

**Use of Maclaurin Series**

The Taylor and Maclaurin series gives a polynomial approximation of a centered function at any point a, while the Maclaurin is always centered on a = 0. Since the behavior of polynomials is easier to understand than functions such as sin(x), we use the Maclaurin series to solve differential equations, infinite sum, and advanced physics calculations. Maclaurin is a subset of the Taylor series.

If we put together some series of infinite items, it would ideally represent a function. The Maclaurin series is just an approximation of a particular function. The series indicates that the accuracy of the function is positively correlated with the number of series.

The number of components in the series is directly related to the order of Maclaurin’s series. The order has the maximum value of n and is expressed by sigma in the formula. The number of components in the series is n +1 because the first term is generated when n = 0. The highest order of the polynomial is n = n.

**Conclusion**

Use this online Maclaurin series calculator to approximate a function for the input values close to zero. It precisely solves the series expansion of the entered function quickly. Our free online calculator generates accurate results for you using the standard formula.

**FAQs**

**How do you convert a function to a Maclaurin series?**

The Maclaurin series allows you to express functions as power series by following these steps:

- Find the first few derivatives of the function until you recognize a pattern.
- Substitute 0 for x into each of these derivatives.
- Plug these values, term by term, into the formula for the Maclaurin series.

**What is meant by the Maclaurin series?**

A Maclaurin series is a power series that allows one to calculate an approximation of a function f(x) for input values close to zero, given that one knows the values of the successive derivatives of the function at zero.

**What is the difference between Maclaurin and Taylor series?**

In the field of mathematics, a Taylor series is defined as the representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. A Maclaurin series is the expansion of the Taylor series of a function about zero.

**What is the purpose of the Taylor and Maclaurin series?**

A Taylor series is an idea used in computer science, calculus, chemistry, physics, and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. There is also a special kind of Taylor series called a Maclaurin series.

**Is the Maclaurin series A special part of the Taylor series?**

The Taylor Series, or Taylor Polynomial, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Maclaurin Polynomial is a special case of the Taylor Polynomial, that uses zero as our single point.