# Maclaurin Series Calculator

For Maclaurin series, x is always zero

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## Maclaurin series calculator

Maclurin series calculator is used to expand the function to make a series around the fixed center point. The point a = 0 is the fixed point in the Maclaurin series. This Maclaurin series solver expands the given function by differentiating it up to the nth order.

## How does the Maclaurin series calculator work?

Maclaurin series expansion calculator is an easy-to-use tool. To expand any function, follow the below steps.

• Enter the function into the input box.
• Press the load example button to use the sample examples.
• Write the order of the function.
• The center point (a=0) is fixed by default.
• Hit the calculate button to get the Maclaurin series of the given function.
• Click the clear button to recalculate.

## What is the Maclaurin series?

A power series that allows one to evaluate an approximation of a function f(x) for input values close to zero, given that one knows the values of the consecutive differentials of the function at a=0 is known as a Maclaurin series. It is a type of the Taylor series.

### The Formula of the Maclaurin series

The general equation or the formula of the Maclaurin series is given below.

$$F\left(x\right)=\sum _{n=0}^{\infty }\frac{f^n\left(0\right)}{n!}\left(x\right)^n$$

• In the equation of Maclaurin series, $$f^n\left(0\right)$$ is the nth derivative of the given function.
• Zero is the fixed center point.
• The total number of terms in the series is “n”.

## How to calculate the Maclaurin series?

Here is an example solved by our Maclaurin series calculator to get the Maclaurin series.

Example

What is the Maclaurin series of sin(x) having n=6?

Solution

Step 1: Take the given data form the problem.

$$f\left(x\right)=sin\left(x\right)$$

$$order=n=6$$

Step 2: Take the general equation of the Maclaurin series for n=6.

$$F\left(x\right)=\sum \:_{n=0}^6\left(\frac{f^n\left(0\right)}{n!}\left(x\right)^n\right)$$

$$F\left(x\right)=\frac{f\left(0\right)}{0!}\left(x\right)^0+\frac{f\:'\left(0\right)}{1!}\left(x\right)^1+\frac{f''\left(0\right)}{2!}\left(x\right)^2+...+\frac{f^{vi}\left(0\right)}{6!}\left(x\right)^6$$ …(1)

Step 3: Now differentiate the given function to get first six derivatives.

$$f\left(x\right)=sin\left(x\right)$$

$$f\:'\left(x\right)=cos\left(x\right)$$

$$f\:''\left(x\right)=-sin\left(x\right)$$

$$f'''\left(x\right)=-cos\left(x\right)$$

$$f^{iv}\left(x\right)=-\left(-sin\left(x\right)\right)=sin\left(x\right)$$

$$f^v\left(x\right)=cos\left(x\right)$$

$$f^{vi}\left(x\right)=-sin\left(x\right)$$

Step 4: Now put a =x=0 in the differentials of sin(x).

$$f\left(0\right)=sin\left(0\right)=0$$

$$f\:'\left(0\right)=cos\left(0\right)=1$$

$$f\:''\left(0\right)=-sin\left(0\right)=-0$$

$$f\:'''\left(0\right)=-cos\left(x\right)=-1$$

$$f^{iv}\left(0\right)=sin\left(x\right)=0$$

$$f^v\left(0\right)=cos\left(0\right)=1$$

$$f^{vi}\left(0\right)=-sin\left(0\right)=0$$

Step 5: Now put the differential values at a=0 in the equation (1).

$$F\left(x\right)=\frac{0}{0!}\left(x\right)^0+\frac{1}{1!}\left(x\right)^1-\frac{0}{2!}\left(x\right)^2-\frac{1}{3!}\left(x\right)^3+\frac{0}{4!}\left(x\right)^4+\frac{1}{5!}\left(x\right)^5-\frac{0}{6!}\left(x\right)^6$$

$$F\left(x\right)=0+\left(x\right)-0-\frac{1}{3!}\left(x\right)^3+0+\frac{1}{5!}\left(x\right)^5-0$$

$$F\left(x\right)=x-\frac{x^3}{6}+\frac{x^5}{120}$$

### References

what is Maclaurin series? | Brilliant Math & Science Wiki. (n.d.)

Maclaurin series formula | Merriam-Webster

Maclaurin series example. Study.com | Take Online Courses. Earn College Credit. Research Schools, Degrees & Careers. (n.d.).