Table of Contents

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.

## Maclaurin Series Calculator:

A Maclaurin series calculator is a tool that calculates the Maclaurin series representation of a given function. The Maclaurin series is a special case of the Taylor series expansion centered at x = 0. For example, if you input the function f(x) = sin(x) into the Maclaurin series calculator, it will calculate the Maclaurin series representation of sin(x), which is the infinite sum of terms involving powers of x.

## Find Maclaurin Series Calculator:

A “find Maclaurin series” calculator helps determine the Maclaurin series expansion of a given function. You can input the specific function, such as f(x) = e^x, and the calculator will calculate and display the corresponding Maclaurin series representation of the function.

## Find The Taylor Series Calculator:

A “find the Taylor series” calculator is a tool that computes the Taylor series expansion of a function around a specific point. By inputting the function and the point of expansion into the calculator, it will calculate the Taylor series representation of the function centered at that point. For example, if you input the function f(x) = ln(x) and the expansion point x = 1, the calculator will provide the Taylor series expansion of ln(x) centered at x = 1.

## Maclaurin Series Calculator With Steps:

A Maclaurin series calculator with steps not only calculates the Maclaurin series representation of a function but also provides a step-by-step explanation of the process. This helps users understand how each term in the series is derived and how the coefficients are determined. The calculator will display the intermediate calculations and the final Maclaurin series expression.

## Taylor Series Calculator With Steps:

A Taylor series calculator with steps calculates the Taylor series expansion of a function and provides a detailed explanation of the steps involved. It shows how the derivatives of the function at the expansion point are used to obtain the coefficients of each term in the series. By following the steps, users can understand the process of deriving the Taylor series representation.

## Maclaurin Series Expansion Calculator:

A Maclaurin series expansion calculator is a tool specifically designed to calculate the Maclaurin series representation of a function. It takes the function as input and performs the necessary calculations to obtain the Maclaurin series expansion. For example, if you input the function f(x) = cos(x) into the calculator, it will calculate the Maclaurin series expansion of cos(x) using the appropriate formula and display the result.

## Power Series Representation Calculator:

A power series representation calculator determines the power series representation of a function. This representation is similar to a Taylor or Maclaurin series but may not be centered at x = 0. By inputting the function into the calculator and specifying the center of the power series, it calculates and displays the power series representation.

## First Four Nonzero Terms Maclaurin Series Calculator:

A calculator for the first four nonzero terms of the Maclaurin series calculates the truncated version of the Maclaurin series representation. Instead of providing the infinite series, it focuses on the first four nonzero terms, which is a finite approximation of the function. For example, if you input the function f(x) = ln(1 + x) into the calculator, it will calculate and display the first four nonzero terms of the Maclaurin series approximation for ln(1 + x).

## Find the Maclaurin Series for f(x):

A calculator to find the Maclaurin series for f(x) calculates the Maclaurin series expansion of a given function. By inputting the specific function, such

as f(x) = 1/(1 – x), it calculates the corresponding Maclaurin series representation for that function.

## Radius of Convergence Maclaurin Series Calculator:

A radius of convergence Maclaurin series calculator determines the radius of convergence for the Maclaurin series expansion of a function. The radius of convergence represents the range of values for which the Maclaurin series converges to the original function. By inputting the function into the calculator, it calculates and provides the radius of convergence, indicating the interval within which the series is valid.

**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.

**How to Find a Maclaurin Series from a Function on a Nspire Graphing Calculator?**

To find a Maclaurin series from a function using a Nspire graphing calculator, you can follow these steps:

a. Enter the function into the calculator. Make sure the function is properly formatted.

b. Access the “Calculus” menu or a similar option on the calculator.

c. Look for the option related to “Series” or “Taylor/Maclaurin series” and select it.

d. Specify the desired order or number of terms for the Maclaurin series.

e. Enter the expansion point or center for the series, usually x = 0 for Maclaurin series.

f. The calculator will calculate and display the Maclaurin series for the entered function up to the specified order or number of terms.

**How to Use a Graphing Calculator to Find the Taylor Polynomial of a Maclaurin Series?**

To use a graphing calculator to find the Taylor polynomial of a Maclaurin series, you can follow these steps:

a. Enter the Maclaurin series into the calculator, ensuring that it is in the proper form and syntax.

b. Access the “Calculus” or “Series” menu on the calculator.

c. Look for an option related to “Taylor Polynomial” and select it.

d. Specify the desired order or number of terms for the Taylor polynomial.

e. The calculator will calculate and display the Taylor polynomial for the Maclaurin series up to the specified order or number of terms.

**How to Find the Exact Sum of a Maclaurin Series Calculator?**

Finding the exact sum of a Maclaurin series often requires an infinite number of terms and is not feasible using a calculator. However, you can use a calculator to approximate the sum up to a certain number of terms. To do this, follow the steps mentioned in the first question to find the Maclaurin series using a calculator. Then, you can evaluate the partial sum by substituting specific values of x into the series expression, summing up the terms, and observing the pattern of convergence.

**How to Do Maclaurin Series with a Calculator?**

To do a Maclaurin series with a calculator, you can follow these general steps:

a. Identify the function for which you want to find the Maclaurin series representation.

b. Determine the number of terms or the desired order of the series.

c. Use the calculator’s built-in functions or specific functions related to series expansions.

d. Enter the function and the desired number of terms or order into the calculator.

e. The calculator will perform the necessary calculations and display the Maclaurin series representation of the function up to the specified order or number of terms.