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**Table of Content**

Maclaurin series calculator helps you to determine the Maclaurin series expansion of a given function around the given points.

Our calculator takes the derivatives for getting the required polynomials that are compulsory and used for getting the series after simplification.

In mathematics, the Maclaurin series is defined as the extended series of specific functions. In this series, the approximated value of the given function can be determined as the sum of the derivatives of any function. When the function expands to zero instead of other values a = 0.

The formula used by the Maclaurin series calculator for computing a series expansion for any function is:

$$ Î£^âˆž_{n=0} \frac{f^n (0)} {n!} x^n $$

Where f^n(0) is the nth order derivative of function f(x) as evaluated and n is the order x = 0. The series will be more precise near the center point. As we shift from the center point a = 0, the series becomes less precise of an approximation of the function.

You can find the expanded series with calculator precisely. But if you want to do it manually, then follow these instructions:

- First, take the function with its range to find the series for f(x).
- The Maclaurin formula is given by \( f(x)=âˆ‘k=0^âˆž f^k (a)* x^k/ k! \)
- Find f^k (a) by evaluating the function derivative and adding the range values in the given function.
- Now, compute the component k! for each step.
- Then, add the obtained values in the formula and apply the sigma function to get the solution.

**Example: **

Calculate Maclaurin expansion of sin(y) up to n = 4?

**Solution:**

Given function f(y)= Sin(y) and order point n = 0 to 4

Maclaurin equation for the function is:

$$ f(y)=âˆ‘k=0^âˆž f (k) (a)* y^k/ k! $$

$$ f(y)â‰ˆ âˆ‘k=0^4 f (k) (a)* y^k/ k! $$

So, calculate the derivative and evaluate them at the given point to get the result into the given formula.

$$ F^0(y) = f (y) = sin(y) $$

Evaluate function:

$$f (0) = 0$$

Take the first derivative \( f^1(y) = [f^0(y)]’ \)

$$ [sin(y)]’ = cos(y) $$

$$[f^0(y)]’ = cos(y)$$

Compute the first derivative

$$(f (0))’ = cos(0) = 1$$

Second Derivative:

$$f^2 (y) = [f^1 (y)]’ = [cos (y)]’ = – sin(y)$$

$$(f (0))â€²â€²=0$$

Now, take the third derivative:

$$ f^3(y) = [f^2(y)]’ = (- sin (y))’ = – cos(y) $$

Calculate the third derivative of \( (f (0))”’ = – cos (0) = -1 \)

Fourth derivative:

$$f^4 (y) = [f^3 (y)]’ = [- cos (y)]’ = sin (y)$$

Then, find the forth derivative of function (f(0))”” = sin(0) = 0

Hence, substitute the values of derivative in the formula

$$ f(y) â‰ˆ 0/0! y^0 + 1/1! y^1 + 0/2! y^2 + (-1)/3! y^3 + 0/4! y^4 $$

$$ f(y) â‰ˆ 0 + x + 0 – 1/6 y^3 + 0 $$

$$ sin(y) â‰ˆ y-1/6 y^3 $$

Maclaurin calculator finds the power series extensions for any function by following these guidelines:

- First, enter the given function with respect to any variable from the drop-down list.
- Now, substitute the value for order n.
- Then, figure out the series and determine the error at that point. (optional)
- Click on the calculate button for the expanded series.

- The calculator computes the series of the function around the given points.
- It takes the derivative of a particular function to obtain the polynomials for getting the final results.
- The Maclaurin polynomial calculator shows step-by-step calculations for all derivatives and polynomials.

From the source of Wikipedia: Analytic functions, Approximation error, and convergence, List of Maclaurin series of some common functions, Exponential function.