## Introduction to Maclaurin Series Calculator:

Maclaurin series Calculator is an amazing digital tool that is used to **find the Maclaurin series** expansion of a given function at a point a=0. It is used to evaluate the approximation value of a given function into an infinite sum of series.

Our tool is beneficial for students, teachers, or researchers who want to get a solution for the Maclaurin series in a few seconds without doing any manual calculations in just one click.

## What is the Maclaurin Series?

The Maclaurin series is defined as the **infinite series** that is generated when you use the differential of a given function at its center where a=0. It is a type of Taylor series at the center point.This type of series is mostly used for functions like sin x,cosx, and e^{x} that give the approximate value of a given function.

Although it does not give as much accuracy in solution as the Taylor series gives it gives a nearly estimated value of x at the origin of a function.

## Formula of Maclaurin Series:

For the function f(x) the **formula** of the Maclaurin series used by the maclaurin polynomial calculator at its origin is given as:

- f(x) is the given function of x variable
- f^n(0) is the nth derivative of a given function f(x) at x=0
- n! is the n factorial value as n=0,1,2,3,...

## How to Calculate the Maclaurin Series?

For **calculating the Maclaurin series** of a function the maclaurin calculator follows different steps. Let's see the calculation of the Maclaurin series calculator with steps:

**Step 1**:

Identify the function f(x) around the origin point a=0

**Step 2**:

Differentiate the given function with respect to x n times

For f(x)=f^{n}(x)

- f′(x)=d/dxf(x)
- f′′(x)=d
^{2}/dx^{2}f(x) - f′′′(x)=d
^{3}/dx^{3}f'(x) - and so on...

So, f`^{(n)}(x) = f^{(n)}(x) n≥0.

**Step 3**:

Now, evaluate these derivative functions at x=0

- f(x) = f(0)
- f′(x) = f'(0)
- f′′(x) = f''(0)
- f′′′(x) = f'''(0)
- and so on...

**Step 4**:

The Maclaurin series expansion of a function f(x) around a=0 is given by:

$$ f(x) \;=\; \sum_{n=0}^{\infty} \frac{f^n (0)}{n!} x^n $$

**Step 5**:

Put the above value into the Maclaurin series formula at x = 0 and simplify it if needed.

## Solved Example of Maclaurin Series:

The solved **example** of the maclaurin series gives you an idea about the working method of the maclaurin polynomial calculator.

### Example: Find formula for nth degree maclaurin polynomial.

$$ f(x) \;=\; e^x $$

**Solution**:

Determine the given function at x=0,

$$ f(x) \;=\; e^x $$

Find the derivative of f(x) with respect to x,

$$ f(x) \;=\; f’(x) \;=\; f’’(x) \;=\; … =\; f^n (x) $$

as f`(x)=e^{x} so that means,

$$ f’(x) \;=\; f’’(x) \;=\; … =\; f^n (x) \;=\; e^x $$

Put x=0 at each function of differentiation,

$$ f(0) \;=\; f’(0) \;=\; f’’(0) \;=\; … =\; f^n(0) \;=\; 1 $$

Put all the above values in the maclaurin series formula,

$$ P_n(x) \;=\; f(a) + f’(a)(x - a) + \frac{f’’ (a)}{2!}(x - a)^2 + \frac{f’’’(a)}{3!}(x - a)^3 + … + \frac{f^n (a)}{n!} (x - a)^n $$

$$ f(0) \;=\; f’(0) \;=\; f’’(0) \;=\; … =\; f^n (0) \;=\; 1 $$

$$ p_0(x) \;=\; f(0) \;=\; 1 $$

$$ p_1(x) \;=\; f(0) + f’(0)x \;=\; 1 + x $$

$$ p_2(x) \;=\; f(0) + f’(0)x + \frac{f’’(0)}{2!}x^2 \;=\; 1 + x + \frac{1}{2}x^2 $$

$$ p_3(x) \;=\; f(0) + f’(0)x + \frac{f’’(0)}{2} x^2 + \frac{f’’’(0)}{3!}x^3 \;=\; 1 + x + \frac{1}{2}x^2 + \frac{1}{3!}x^3 $$

$$ p_n(x) \;=\; f(0) + f’(0)x + \frac{f’’(0)}{2}x^2 + \frac{f’’’(0)}{3!}x^3 + … + \frac{f^n(0)}{n!}x^n $$

$$ =\; 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + … + \frac{x^n}{n!} $$

$$ =\; \sum_{k=0}^n \frac{x^k}{k!} $$

The solution of a given function f(x)=e^{x} is,

$$ =\; \sum_{k=0}^{n} \frac{x^k}{k!} $$

The maclaurin series function graph is,

### PASTE THE GRAPH HERE!

## How to Use Maclaurin Series Calculator?

Maclaurin calculator has a user-friendly interface that provides you with a given function at its center point.

You just need to follow some instructions that are given below before using this maclaurin polynomial calculator. The uses are given below,

- Enter your given function in the respective field which maclaurin series expansion solution you want.
- Add the orders of derivation n in the relevant box.
**Choose the variable**which you want to differentiate the given function for the maclaurin series.- Click on the calculate button to get the result in the form of expansion of the maclaurin series.
- The recalculate button will bring you back to the home page for more calculation of maclaurin series examples.

## Final Result of Maclaurin Expansion Calculator:

Maclaurin series expansion calculator will get the **result** of the given function instantly after adding the input value which may include as:

**Solution section**:

It will provide you with the solution of given maclaurin series problem.

**Possible steps section**:

It gives you the Maclaurin series expansion solution in the stepwise method.

- The Recalculate button provides a new page for further evaluation of the Maclaurin series.

## Benefits of Maclaurin Calculator:

The maclaurin polynomial calculator will provide you with **multiple benefits** as it will give solutions of various types of the Maclaurin series function. These benefits are:

- The Mclaurin series calculator gives you an accurate solution of the Maclaurin problem without any man made error because it has up to date features.
- You can use this tool to solve different examples of maclaurin series problems so that you get a strong hold on this concept.
- Maclaurin series calculator saves time that you consume in doing manual calculation of the Maclaurin series function.
- It has a simple design that allows everyone to easily use it to solve the Maclaurin series problem.
- Find the maclaurin series is a manageable tool as it is used from any electronic device through the internet.
- Our maclaurin expansion calculator provides a speedy calculation of the Maclaurin function in fraction of second without any mistake.