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.
To find the Maclaurin series of functions, follow the below steps.
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:
\(F\left(x\right)=\sum _{n=0}^{\infty }\frac{f^n\left(0\right)}{n!}\left(x\right)^n\)
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 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.
\( f\left(0\right)=cos\left(0\right)=1\)
\( f'\left(0\right)=-sin\left(0\right)=0\)
\(f''\left(0\right)=-cos\left(0\right)=-1\)
\( f’’’\left(0\right)=-\left(-sin\left(0\right)\right)=sin\left(0\right)=0\)
\( f^{iv}\left(0\right)=cos\left(0\right)=1\)
\( f^{v}\left(0\right)=-sin\left(0\right)=0\)
\( f^{vi}\left(0\right)=-cos\left(0\right)=-1\)
\( f^{vii}\left(0\right)=-\left(-sin\left(0\right)\right)=sin\left(0\right)=0\)
Step 4: Put the derivatives of cos(x) in (1).
\( F\left(x\right)=\frac{1}{0!}\left(x\right)^0+\frac{0}{1!}\left(x\right)^1-\frac{1}{2!}\left(x\right)^2+\frac{0}{3!}\left(x\right)^3+\frac{1}{4!}\left(x\right)^4+\frac{0}{5!}\left(x\right)^5-\frac{1}{6!}\left(x\right)^6+\frac{0}{7!}\left(x\right)^7\)
\( F\left(x\right)=1+0-\frac{1}{2}\left(x\right)^2+0+\frac{1}{24}\left(x\right)^4+0-\frac{1}{720}\left(x\right)^6+0\)
\(F\left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}\)
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) | \(x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5+\ldots \:\) |
