Unformatted text preview: x = a is the polynomial
DE Math 128 318 (B. Forrest)2 CHAPTER 4. Sequences and Series n Tn,a (x) =
k=0 4.5. Introduction to Taylor Series f k (a)
(x − a)k
k! = f (a) + f (a)(x − a) + f (n) (a)
f (a)
(x − a)2 + · · · +
(x − a)n
2!
n! The remarkable thing about Tn,a (x) is that for any k between 0 and n,
(k )
Tn,a (a) = f (k) (a). That is Tn,a encodes not only the value of f (x) at x = a but all of its ﬁrst n derivatives as
well. Moreover, this is the only polynomial of degree n or less that does so.
Example.
Find all of the Taylor polynomials up to degree 5 for the function f (x) = cos(x) with center
x = 0.
We have already seen that f (0) = cos(0) = 1, f (0) = − sin(0) = 0 and f (0) = − cos(0) =
−1. It follows that
T0,0 (x) = 1,
and
T1,0 (x) = L0 (x) = 1 + 0(x − 0) = 1
for all x, while
T2,0 = 1 + 0(x − 0) + x2
−1
(x − 0)2 = 1 − .
2!
2 Since f (x) = sin(x), f (4) (x) = cos(x), and f (5) (x) = − sin(x), we get f (0) = sin(0) = 0,
f (4) (0) = cos(0) = 1 and f (5) (x) = − sin(0) =...
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 Spring '09
 Math, Derivative, Power Series, Taylor Series, Sequences And Series, Taylor's theorem, B. Forrest

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