Math Calculators ▶ Maclaurin Series Calculator
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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:
$$ Σ^∞_{n=0} \frac{f^n (0)} {n!} x^n $$
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.
You can find the expanded series with our Maclaurin series calculator precisely. But if you want to do it manually, then follow these instructions:
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
$$ f(y) ≈ 0/0! y^0 + 1/1! y^1 + 0/2! y^2 + (-1)/3! y^3 + 0/4! y^4 $$
$$ f(y) ≈ 0 + x + 0 – 1/6 y^3 + 0 $$
$$ sin(y) ≈ y-1/6 y^3 $$
Maclaurin calculator finds the power series extensions for any function by following these guidelines:
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.