Consider the functions C: R → R and S: R → R defined by 00 C(x) = Σ n=0 2n (-1)" x ² (2n)! Use the Power Series to show that C'(x) C(0) = 1 and S(0) = 0 and S(x) == = Σ n=0 n 2n+1 (2n+1)! S(x) and S(x) = C(x) for all x & R and that
Consider the functions C: R → R and S: R → R defined by 00 C(x) = Σ n=0 2n (-1)" x ² (2n)! Use the Power Series to show that C'(x) C(0) = 1 and S(0) = 0 and S(x) == = Σ n=0 n 2n+1 (2n+1)! S(x) and S(x) = C(x) for all x & R and that
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Consider the functions C: R→ R and S: R → R defined by
C(x)
=
Σ
n=0
n 2n
(−1)"x²
(2n)!
Use the Power Series to show that C'(x)
C(0) = 1 and S(0) = 0
and S(x)
=
00
Σ
n=0
n 2n+1
(-1)"x²
(2n+1)!
=-
- S(x) and S(x) = C(x) for all x & R and that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F22dbf9f2-e64d-4262-a1a1-46b878e1b1e8%2Flgvk8qd_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the functions C: R→ R and S: R → R defined by
C(x)
=
Σ
n=0
n 2n
(−1)"x²
(2n)!
Use the Power Series to show that C'(x)
C(0) = 1 and S(0) = 0
and S(x)
=
00
Σ
n=0
n 2n+1
(-1)"x²
(2n+1)!
=-
- S(x) and S(x) = C(x) for all x & R and that
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