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Maclaurin Series Examples

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. The 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.


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.


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


Step 1: Write the given terms.


Order = n = 7

Fixed point = a = 0

Step 2: Write the equation of the 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.

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

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


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

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

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

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

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

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

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

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


Table of some examples of Maclaurin series

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

Maclaurin series forResult
sinx\(x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7+\frac{1}{9!}x^9+\ldots \)
arctanx\(x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\frac{1}{9}x^9+\ldots \)
cos(x^2)\(1-\frac{1}{2}x^4+\frac{1}{24}x^8+\ldots \)
ln(1+x)\(x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5+\ldots \:\)


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