Taylor Series - Exercise 1 Solution:
$$f(x) = \sum_{n=0}^{\infty} \frac{x^n f^{(n)}(0)}{n!} = f(0) + x f^{(1)}(0) + \frac{x^2}{2!} f^{(2)}(0) + \dots$$
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:
$$f(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$$
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