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Maclaurin Series Calculator + Online Solver With Free Steps

The Maclaurin series calculator is a free online tool for expanding the function around a fixed point. In the Maclaurin series, the center point is set at a = 0. It determines the series by taking the function’s derivatives up to order n.

What Is a Maclaurin Series Calculator?

The Maclaurin series calculator is a free online tool for expanding the function around a fixed point. A Maclaurin series is a subset of the Taylor series. A Taylor series gives us a polynomial approximation of a function with a center at point a, but a Maclaurin series is always centered on a = 0.

A Maclaurin series can be used to assist in the solution of differential equations, infinite sums, and complex physics issues since the behavior of polynomials can be simpler to comprehend than functions like sin(x). The function will be perfectly represented by a Maclaurin series with infinite terms.

A finite Maclaurin series is only a rough approximation of the function, and the number of terms in the series has a positive correlation with how accurately it approximates the function. We can obtain a more precise illustration of the function by running additional terms of a Maclaurin series.

The Maclaurin series’ degree is directly correlated with the number of words in the series. The formula shown below uses sigma notation to represent the largest n value, which is the degree. Since the first term is generated with n = 0, the total number of terms in the series is n + 1. n = n is the polynomial’s highest power.

How To Use a Maclaurin Series Calculator

You can use the Maclaurin Series Calculator by following the detailed guidelines given below, and the calculator will provide the desired results in just a moment. Follow the instructions to get the value of the variable for the given equation.

Step 1

Fill in the appropriate input box with two functions.

Step 2

Click on the “SUBMIT” button to determine the series for a given function and also the whole step-by-step solution for the Maclaurin Series Calculator will be displayed.

How Does Maclaurin Series Calculator Work?

The calculator works by finding the sum of the given series using the concept of Maclaurin Series. The extended series of certain functions is referred to as the Maclaurin series in mathematics.

The sum of any function’s derivatives in this series can be used to calculate the approximate value of the provided function. When a = 0, the function expands to zero rather than any other values.

Maclaurin Series Formula

The Maclaurin series calculator uses the following formula to determine a series expansion for any function:

\[\sum_{n=0}^{\infty} \frac{f^n (0)} {n!} x^n\]

Where n is the order x = 0 and $f^n(0)$ is the nth order derivative of the function f(x) as evaluated. Near the centroid, the series will get more precise. The series becomes less exact as we move away from the center point a = 0.

Use of Maclaurin Series

The Taylor and Maclaurin series approximate a centered function with a polynomial at any point a, whereas the Maclaurin is uniformly focused on a = 0.

We utilize the Maclaurin series to solve differential equations, infinite sums, and complex physics calculations because the behavior of polynomials is simpler to understand than functions like sin (x).

The Taylor series includes the Maclaurin as a subset. The ideal representation of a function would be a set of infinite elements. The Maclaurin series only approximates a specific function.

The series shows a positive correlation between the number of series and the correctness of the function. The order of Maclaurin’s series is closely correlated with the number of components in the series. The formula’s sigma is used to represent the order, which has the highest possible value of n.

Because the first term is formed when n = 0, the series has n + 1 components. The polynomial has an order of n = n.

Steps for Locating the Maclaurin Series of Function

This Maclaurin series calculator accurately calculates the expanded series, but if you prefer to do it by hand, then adhere to these guidelines:

  • To find the series for f(x), start by taking the function with its range.
  • The formula for Maclaurin is provided by \[ f(x)= \sum_{k=0}^{\infty} f^k (a) \cdot \frac{x^k}{k!}\]
  • By calculating the derivative of the given function and combining the range values, one can determine $ f^k (a) $.
  • Now, calculate the step’s component, k!
  • To find the solution, add the calculated values to the formula and use the sigma function.

Solved Examples

Let’s explore some examples to better understand the Maclaurin Series.

Example 1

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:

\[ f(y)= \sum_{k=0}^{\infty}  f (k) (a) \cdot \frac{y^k}{ k!} \]

\[ f(y) \approx   \sum_{k=0}^{4}  f (k) (a) \cdot \frac{y^k}{ k!} \]

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))”= 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 fourth derivative of function (f(0))”” = sin(0) = 0 

Hence, substitute the values of derivative in the formula

\[ f(y) \approx  \frac{0}{0!} y^0 + \frac{1}{1!} y^1 + \frac{0}{2!} y^2 + \frac{ (-1)}{3!} y^3 + \frac{0}{4!} y^4 \]

\[ f(y) \approx 0 + x + 0 – \frac{1}{6} y^3 + 0 \]

\[ \sin (y) \approx y  –  \frac{1}{6} y^3 \]

Example 2

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

Solution:

Write the given terms.

f(x) = cos (x) 

Order = n = 7

Fixed Point = a = 0

Writing the equation of the Maclaurin series for n =7.

\[ F(x) = \sum_{n=0}^{7} (\frac{f^n(0)}{n!}(x)^n) \]

\[ F(x) = \frac{f(0)}{0!}(x)^0)+ \frac{f’(0)}{1!}(x)^1)+ \frac{f”(0)}{2!}(x)^2)+ … + \frac{f^7(0)}{7!}(x)^7)\]

Now calculating the first seven derivatives of cos(x) at x=a=0.

f(0) = cos (0) = 1 

f’(0) = -sin (0) = 0 

f”(0) = -cos (0) = -1 

f”'(0) = -(-sin (0)) = 0 

$f^4(0) $= cos (0) = 1 

$f^5(0)$ = -sin (0) = 0 

$f^6(0)$ = -cos (0) = -1 

$f^7(0) $= -(-sin (0)) = 0 

\[ F(x) = \frac{1}{0!}(x)^0+ \frac{0}{1!}(x)^1 – \frac{1}{2!}(x)^2 + \frac{0}{3!}(x)^3 +\frac{1}{4!}(x)^4 + \frac{0}{5!}(x)^5 – \frac{1}{6!}(x)^6 + \frac{0}{7!}(x)^7 \]

\[ F(x) = 1 – \frac{x^2}{2}+ \frac{x^4}{24} – \frac{x^6}{720} \]

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