Table of derivatives for \( f(x) = \cos(x) \) at \( x = \pi/2 \):

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:

$$\text{ANSWER: } \sum_{k=1}^{\infty} \frac{(-1)^k (x - \pi/2)^{2k-1}}{(2k-1)!}$$


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