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Maclaurin Series Calculator

Enter a function:

W.R.T:

Order n:

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

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Maclaurin Series Calculator

The Maclaurin series calculator helps determine the Maclaurin series expansion of a function around a specified point (usually 0). It computes the derivatives required for the polynomial terms and simplifies them to provide the series.

What is Maclaurin Series?

In mathematics, the Maclaurin series is a special case of the Taylor series where the expansion is centered at a = 0. It approximates a function as the sum of its derivatives at zero:

Maclaurin Series Formula:

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

Where:

f⁽ⁿ⁾(0) = nth derivative of f(x) evaluated at 0
n! = factorial of n

This series provides a good approximation near x = 0, but becomes less precise as you move further from zero.

Steps to Find the Maclaurin Series Manually

Start with the function f(x) you want to expand.
Use the formula: $$ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0) x^k}{k!} $$
Compute each derivative f⁽ᵏ⁾(x) and evaluate it at x = 0.
Calculate k! for each term.
Substitute the values into the series and sum them to get the approximation.

Example: Maclaurin Series of sin(y) up to n = 4

Given: f(y) = sin(y), n = 0 to 4

Maclaurin series formula:

$$ f(y) \approx \sum_{k=0}^{4} \frac{f^{(k)}(0) y^k}{k!} $$

Step 1: Compute Derivatives

f⁰(y) = sin(y), f⁰(0) = 0
f¹(y) = cos(y), f¹(0) = 1
f²(y) = -sin(y), f²(0) = 0
f³(y) = -cos(y), f³(0) = -1
f⁴(y) = sin(y), f⁴(0) = 0

Step 2: Substitute into the formula

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

$$ f(y) \approx 0 + y + 0 - \frac{1}{6} y^3 + 0 $$

$$ \sin(y) \approx y - \frac{1}{6} y^3 $$

How the Maclaurin Series Calculator Works

Input:

Enter the function with respect to any variable from the dropdown list.
Specify the order n (number of terms in the series).
(Optional) Specify a point to calculate error or approximation.
Click Calculate to generate the series.

Output:

The calculator provides the Maclaurin series expansion of the function.
Automatically computes derivatives and factorial terms for all powers.
Displays step-by-step calculations for better understanding.

Reference

For more details on Maclaurin series, derivatives, and convergence, see Wikipedia: List of Maclaurin series of common functions.

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