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.).

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