Maclaurin Series

What is the Maclaurin Series?

The Maclaurin series is a power series expansion of a function $f(x)$ centered at $x = 0$. It is a special case of the Taylor series with $a = 0$.

Common Expansions

$e^x$

Since $f^{(n)}(x) = e^x$, we have $f^{(n)}(0) = 1$ for all $n$:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$\sin x$

Only odd-degree terms survive:

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

$\cos x$

Only even-degree terms survive:

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

$\frac{1}{1-x}$ (Geometric series)

$$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \quad (|x| < 1)$$

Connection to Euler’s Formula

Substituting $x = i\theta$ into the expansions of $e^x$, $\sin x$, and $\cos x$ yields:

$$e^{i\theta} = \cos\theta + i\sin\theta$$

This is Euler’s formula.