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**Unformatted text preview: **rom the Approximation Theorem
0
that we can ﬁnd a constant M1 such that for any u ∈ [−1, 1]
−M1 u2 ≤ eu − (1 + u) ≤ M1 u2
DE Math 128 333 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series
4 since 1 + u is the ﬁrst degree Taylor polynomial of eu . Now if x ∈ [−1, 1], then u = x ∈
2
4
1
[−1, 1]. In fact, u ∈ [0, 2 ). It follows that if x ∈ [−1, 1] and we substitute u = x , then we
2
get
x4
x4
−M1 x8
−M1 x8
≤ e 2 − (1 + ) ≤
4
2
4
We also can show that there exists a constant M2 such that for any v ∈ [−1, 1]
−M2 v 4 ≤ cos(v ) − (1 −
since 1 − v2
2 v2
) ≤ M2 v 4
2 is the third degree Taylor polynomial for cos(v ). If x ∈ [−1, −1] then so is x2 . If we let v = x2 , then we see that
−M2 x8 ≤ cos(x2 ) − (1 − x4
) ≤ M2 x 8
2 The next step is to multiply each term in the previous inequality by −1 to get
−M2 x8 ≤ (1 − x4
) − cos(x2 ) ≤ M2 x8 .
2 (Remember, multiplying by a negative number reverses the inequality)
We can now add ou...

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