1. Taylor Series Expansion
The Taylor Series is a powerful method to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point.
The general Taylor Series of a function \( f(x) \) about a point \( a \) is:
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + ...
2. Maclaurin Series
The Maclaurin Series is a special case of the Taylor Series where the expansion is around \( a = 0 \).
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...
Examples:
e^x = 1 + x + x²/2! + x³/3! + ...sin(x) = x - x³/3! + x⁵/5! - ...cos(x) = 1 - x²/2! + x⁴/4! - ...
3. Approximating Function Values
Taylor and Maclaurin series are often used to approximate function values by taking a few leading terms of the series.
Example: Approximate \( e^{0.1} \) using 3 terms of the Maclaurin series:
e^x ≈ 1 + x + x²/2 = 1 + 0.1 + (0.1)²/2 = 1.105
The more terms we include, the better the approximation. This is particularly useful in numerical methods and calculators.