| n | \( f^{(n)}(x) \) | \( f^{(n)}(\pi/2) \) |
|---|---|---|
| 0 | \( \cos(x) \) | 0 |
| 1 | \( -\sin(x) \) | -1 |
| 2 | \( -\cos(x) \) | 0 |
| 3 | \( \sin(x) \) | 1 |
| 4 | \( \cos(x) \) | 0 |
| 5 | \( -\sin(x) \) | -1 |
| 6 | \( -\cos(x) \) | 0 |
| 7 | \( \sin(x) \) | 1 |
The table shows that terms with even \( n \) are zero. Nonzero terms alternate between -1 and 1. To capture this pattern, use \( n = 2k - 1 \) for \( k = 1, 2, 3, \dots \).
General formula:
$$f(x) = \sum_{n=0}^{\infty} \frac{(x - \pi/2)^n}{n!} f^{(n)}(\pi/2)$$
Substitute nonzero terms:
$$\text{ANSWER: } \sum_{k=1}^{\infty} \frac{(-1)^k (x - \pi/2)^{2k-1}}{(2k-1)!}$$