Understanding Maclaurin Series

The Maclaurin series is a special case of Taylor series expansion around zero. It's widely used in physics and engineering to approximate complex functions with polynomials, enabling simplifications in calculations and analysis.

Step-by-Step Calculation

Start with a function f(x) that is infinitely differentiable at x=0
Compute derivatives f'(x), f''(x), ..., fⁿ(x)
Evaluate each derivative at x=0: f(0), f'(0), f''(0), ...
Construct the series: f(0) + f'(0)x + f''(0)x²/2! + ... + fⁿ(0)xⁿ/n!

Common Examples

Function	Maclaurin Series
eˣ	1 + x + x²/2! + x³/3! + ...
sin(x)	x - x³/3! + x⁵/5! - ...
cos(x)	1 - x²/2! + x⁴/4! - ...

Important Notes

• The series converges only within its radius of convergence

• Higher degrees improve approximation accuracy

• Some functions require complex analysis for full expansion

Maclaurin Series Calculator

Maclaurin Series

Understanding Maclaurin Series

Step-by-Step Calculation

Common Examples

Important Notes

Derivative Calculator

Frequently Asked Questions

What is the difference between Taylor and Maclaurin series?

Which functions can be expanded into Maclaurin series?

How many terms should I use for a good approximation?

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Maclaurin Series Calculator - Series at Zero

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