The Maclaurin Series
A Maclaurin Series is just a Taylor series where the starting point is zero (\(a=0\)). The most important Maclaurin series in science is for \(e^x\).
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
Worked Examples
Example 1: Calculating 'e'
Set \(x=1\). Then \(e = 1 + 1 + 1/2 + 1/6 + 1/24 + \dots \approx 2.718\).
Example 2: Small x Approximation
If \(x\) is very small (like \(0.001\)), then \(e^x \approx 1 + x\). All the other terms are so small they can be ignored. This is used in physics constantly.
The Bridge to Quantum Mechanics
This series expansion of \(e^x\) is how we define what it means to "exponentiate" a matrix or an operator. In Quantum Mechanics, the time-evolution operator is \(U = e^{-iHt/\hbar}\). Since \(H\) is an operator (not a number), we can only calculate this by using the Maclaurin series: \(U = I - iHt/\hbar + \dots\). This expansion is how we calculate the first-order corrections to a particle's energy. It is the fundamental link between continuous time and discrete quantum states.