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Maclaurin Series Calculator

Maclaurin Series Calculator

Enter a function:

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Order n:

Calculate the series and determine the error at that point (optional):

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An online Maclaurin series calculator helps you to determine the Maclaurin series expansion of a given function around the given points. This calculator takes the derivatives for getting the required polynomials that are compulsory and used for getting the series after simplification. In this context, you can find more about series such as how to find Maclaurin series with its formula and examples.

What is Maclaurin Series?

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.

Maclaurin Series Formula:

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

$$ Σ^∞_{n=0} \frac{f^n (0)} {n!} x^n $$

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.

However, an online Arithmetic Sequence Calculator that helps you to calculate the Arithmetic sequence, nth value, and sum of the arithmetic sequence.

Use of Maclaurin Series:

The Taylor and Maclaurin series gives a polynomial approximation of a centered function at any point a, while the Maclaurin is always centered on a = 0. Since the behavior of polynomials is easier to understand than functions such as sin(x), we use the Maclaurin series to solve differential equations, infinite sum, and advanced physics calculations. Maclaurin is a subset of the Taylor series. If we put together some series of infinite items, it would ideally represent a function. The Maclaurin series is just an approximation of a particular function. The series indicates that the accuracy of the function is positively correlated with the number of series.

The number of components in the series is directly related to the order of Maclaurin’s series. The order has the maximum value of n and is expressed by sigma in the formula. The number of components in the series is n +1 because the first term is generated when n = 0. The highest order of the polynomial is n = n.

Finding Maclaurin Series of Function with steps:

You can find the expanded series with our Maclaurin series 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 \( f(x)=∑k=0^∞ f^k (a)* x^k/ k! \)
  • 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.

However, an Online Factorial Calculator allows computing the factorial of a given n positive number.

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:

$$ f(y)=∑k=0^∞ f (k) (a)* y^k/ k! $$

$$ f(y)≈ ∑k=0^4 f (k) (a)* y^k/ k! $$

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:

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

$$(f (0))′′=0$$

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

$$ f(y) ≈ 0/0! y^0 + 1/1! y^1 + 0/2! y^2 + (-1)/3! y^3 + 0/4! y^4 $$

$$ f(y) ≈ 0 + x + 0 – 1/6 y^3 + 0 $$

$$ sin(y) ≈ y-1/6 y^3 $$

How Maclaurin Series Calculator Works?

An online maclaurin calculator finds the power series extentions for any function by following these guidelines:

Input:

  • 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.

Output:

  • The Maclaurin series 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.

FAQ:

Is the Maclaurin series converging?

Since the limit is 0, the series converges with the alternating series test that means the Maclaurin series converges at the left endpoint of the interval x.

What is the Maclaurin series for cos x?

The Maclaurin series of cos (x) is only the Taylor series of cos (x) at the point x = 0. If we want to compute the series expansion for any value of x, we can consider several techniques.

When can you use the Maclaurin series?

A Maclaurin series can be used to estimate the functions, determine the anti-derivative of a complex function, or evaluate the sum that would otherwise be impossible to calculate.

Conclusion:

Use this online Maclaurin series calculator to approximate a function for the input values close to zero. It precisely solves the series expansion of the entered function quickly. Our free online calculator generates accurate results for you using the standard formula.

Reference:

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

From the source of Krista King Math: Maclaurin series as the Taylor series centered around a=0, How to build the Maclaurin series, Finding the nth-degree Maclaurin series.

From the source of Brilliant: Derivation, Interval, and Radius of Convergence, Frequently Used Maclaurin Series.