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## Maclaurin Series Calculator

Maclaurin series calculator is an online tool used to expand the function around the fixed point. The center point is fixed in the Maclaurin series as a = 0. It calculates the series by taking the derivatives of the function up to order n.

## How does this Maclaurin polynomial calculator work?

To find the Maclaurin series of functions, follow the below steps.

• Write the one variable function into the input box.
• Write the nth order of the series.
• The center point is fixed by default. Default value is a = 0.
• Press the calculate button to get the result.
• To enter a new function, press the reset button.

## What is the Maclaurin series?

Maclaurin series is a form of Taylor series in which the center point is always fixed as a = 0. In the Taylor series, we can choose any value of a but in the Maclaurin series, the point is a=0 always.

### Formula

The formula for the Maclaurin series is:

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

• $$f^n\left(0\right)$$ is the nth derivative of the function.
• 0 is the fixed point.
• “n” is the total number.

## How to calculate the Maclaurin series?

Following is an example of the Maclaurin series.

Example

Calculate the Maclaurin series of cos(x) up to order 7.

Solution

Step 1: Write the given terms.

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

Order = n = 7

Fixed point = a = 0

Step 2: Write the equation of Maclaurin series for n=7.

$$F\left(x\right)=\sum _{n=0}^7\left(\frac{f^n\left(0\right)}{n!}\left(x\right)^n\right)$$

$$F\left(x\right)=\frac{f\left(0\right)}{0!}\left(x\right)^0+\frac{f\:'\left(0\right)}{1!}\left(x\right)^1+\frac{f\:''\left(0\right)}{2!}\left(x\right)^2+...+\frac{f^{vii}\left(0\right)}{7!}\left(x\right)^7$$ …(1)

Step 3: Now calculate the first seven derivatives of cos(x) at x=a=0.

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

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

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

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

$$f^{iv}\left(0\right)=cos\left(0\right)=1$$

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

$$f^{vi}\left(0\right)=-cos\left(0\right)=-1$$

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

Step 4: Put the derivatives of cos(x) in (1).

$$F\left(x\right)=\frac{1}{0!}\left(x\right)^0+\frac{0}{1!}\left(x\right)^1-\frac{1}{2!}\left(x\right)^2+\frac{0}{3!}\left(x\right)^3+\frac{1}{4!}\left(x\right)^4+\frac{0}{5!}\left(x\right)^5-\frac{1}{6!}\left(x\right)^6+\frac{0}{7!}\left(x\right)^7$$

$$F\left(x\right)=1+0-\frac{1}{2}\left(x\right)^2+0+\frac{1}{24}\left(x\right)^4+0-\frac{1}{720}\left(x\right)^6+0$$

$$F\left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}$$

## Table of some examples of Maclaurin series

Here are some examples and results of the Maclaurin series solved by our Maclaurin calculator.

 Maclaurin series for Result e^x $$1+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4+\ldots$$ sinx $$x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7+\frac{1}{9!}x^9+\ldots$$ arctanx $$x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\frac{1}{9}x^9+\ldots$$ sin^2x $$x^2-\frac{1}{3}x^4+\frac{2}{45}x^6-\frac{1}{315}x^8+\frac{2}{14175}x^{10}+\ldots$$ cos(x^2) $$1-\frac{1}{2}x^4+\frac{1}{24}x^8+\ldots$$ ln(1+x) $$x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5+\ldots \:$$

## References

Use android or iOS app of our limit calculator on your mobile 