$$f(x) = \sum_{n=0}^{\infty} \frac{x^n f^{(n)}(0)}{n!} = f(0) + x f^{(1)}(0) + \frac{x^2}{2!} f^{(2)}(0) + \dots$$
We need to determine \( f^{(n)}(0) \) given \( f(x) = \cos x \).
| n | \( f^{(n)}(x) \) | \( f^{(n)}(0) \) |
|---|---|---|
| 0 | \( \cos x \) | \( \cos 0 = 1 \) |
| 1 | \( -\sin x \) | \( -\sin 0 = 0 \) |
| 2 | \( -\cos x \) | \( -1 \) |
| 3 | \( \sin x \) | \( 0 \) |
| 4 | \( \cos x \) | \( 1 \) |
Using the table:
$$f(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$$