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**Unformatted text preview: **it by T3,a (x).
Given a function f (x), we will also write
T0,a (x) = f (a)
and
T1,a (x) = La (x) = f (a) + f (a)(x − a)
and call these polynomials the zero-th degree and ﬁrst degree Taylor polynomials of f (x)
centered at x = a, respectively.
Observe that with the convention that 0! = 1! = 1 and (x − a)0 = 1 , we have the following
f (a)
(x − a)0
0!
f (a)
f (a)
T1,a (x) =
(x − a)0 +
(x − a)1
0!
1!
f (a)
f (a)
f (a)
T2,a (x) =
(x − a)0 +
(x − a)1 +
(x − a)2
0!
1!
2!
f (a)
f (a)
f (a)
f (a)
T3,a (x) =
(x − a)0 +
(x − a)1 +
(x − a)2 +
(x − a)3
0!
1!
2!
3! T0,a (x) = Recall that f (k) (a) denotes the k -th derivative of f (x) at x = a and that by convention
f (0) (x) = f (x). Then using summation notation, we get
0 T0,a (x) =
k=0
1 T1,a (x) =
k=0
2 T2,a (x) =
k=0 f k (a)
(x − a)k
k!
f k (a)
(x − a)k
k!
f k (a)
(x − a)k
k! and
3 T3,a (x) =
k=0 f k (a)
(x − a)k
k! This leads us to the following deﬁnition:
Deﬁnition.
Assume that f (x) is n-times diﬀerentiable at x = a. The n-th degree Taylor polynomial for
f (x) centered at...

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