Taylor Series

Forrest2 chapter 4 sequences and series 45

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Unformatted text preview: imit: Theorem. Let x0 ∈ R and let M > 0 be any constant. Then M | x0 |k =0 k→∞ k! lim Example. Let f (x) = cos(x) and a = 0. Let x0 be any point in R. Taylor’s Theorem shows that for each k there exists a point ck between 0 and x0 such that | Rk,a (x0 ) |=| f (k+1) (ck ) k+1 x | (k + 1)! 0 We have seen that if f (x) = cos(x), then f (x) = − sin(x), f (x) = − cos(x), f (x) = sin(x) and f (4) (x) = cos(x). Since the fourth derivative is again cos(x), the 5-th, 6th, 7-th and 8-th derivative will be, respectively, f (5) (x) = − sin(x), f (6) (x) = − cos(x), f (7) (x) = sin(x) and f (8) (x) = cos(x). The pattern will then be repeated for the 9-th, 10-th, 11-th and 12-th derivatives, and then for every group of four derivatives thereafter. In fact, what we have just shown is that if f (x) = cos(x), then for any k f (k) (x) = if k = 4j cos(x) − sin(x) if k = 4j + 1 . − cos(x) if k = 4j + 2 sin(x) if k = 4j + 3 where j = 0, 1, 2, · · · . However, this means that no matter what k is and no matter where ck...
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