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**Unformatted text preview: **for π .
Example. Find the Taylor series centered at x = 0 for the integral function
x cos(t2 ) dt. F (x) =
0 Also ﬁnd F (9) (0) and F (16) (0).
We know that for any u ∈ R,
∞ (−1)n cos(u) =
n=0 x 2n
.
(2n)! If we let u = t2 , we get that for any t ∈ R,
∞ 2 2n
n (t ) (−1) cos(u) =
n=0 (2n)! ∞ (−1)n =
n=0 t4n
.
(2n)! The Integration Theorem for Power Series gives us that
x cos(t2 ) dt F (x) =
0 x∞ (−1)n
0 t4n
dt
(2n)! n=0 ∞ t4n
dt
(2n)! (−1)n =
x =
n=0
∞ 0 [(−1)n =
n=0
∞ (−1)n =
n=0 t4n+1
|x ]
(4n + 1)(2n)! 0 x4n+1
.
(4n + 1)(2n)! This is valid for any x ∈ R. Moreover, by the Uniqueness Theorem for Power Series
Representations, this must be the Taylor series for F (x).
DE Math 128 340 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series To ﬁnd F (9) (0), we recall that if
∞ ak x k , F (x) =
k=0 then
a9 = F (9) (0)
.
9! This tells us that to ﬁnd F (9) (0) we must ﬁrst identify the coeﬃcient of x9 in
∞ [(−1)n
n=0 x4n+1
.
(4n + 1)(2n)! It is an easy observation that to get x9 we let n = 2...

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