Taylor & Maclaurin Series
Table of Contents
Taylor Series #
The Taylor series is a power series representation of a function. It is a way to approximate a function as an infinite sum of terms. The Taylor series is centered around a point \( \small a \) and is given by the summation:
$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} \cdot (x - a)^n $$
Where
- \( \small f(x) \) is the function being approximated,
- \( \small f^{(n)}(a) \) denotes the \( \small n \)th derivative of the function evaluated at \( \small x = a \),
- \( \small a \) is the point around which the series is centered, and
- \( \small n! \) is the factorial of \( \small n \)
Maclaurin Series #
The Maclaurin series is a special case of the Taylor series where the series is centered around \( \small a = 0 \). The Maclaurin series is then given by the summation:
$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} \cdot x^n $$