20. 3 5 Recall the power series expansion sin(x) = x - + 3! 5! 71 Use the given expansion to derive a power series expansion for sin(x²). Show at least 4 non-zero terms. i. ii. iii. iv. + Use the given expansion to derive a power series expansion forf sin(x²) dx. Show at least 4 non-zero terms. Use the Alternating Series Estimation Theorem to determine an upper bound for the error in using only the first 2 terms to approximate sin(x²) dx. Use the first 3 term in the expansion in part (ii) to approximate sin(x²) dx. Round your answer to 6 decimal places.
20. 3 5 Recall the power series expansion sin(x) = x - + 3! 5! 71 Use the given expansion to derive a power series expansion for sin(x²). Show at least 4 non-zero terms. i. ii. iii. iv. + Use the given expansion to derive a power series expansion forf sin(x²) dx. Show at least 4 non-zero terms. Use the Alternating Series Estimation Theorem to determine an upper bound for the error in using only the first 2 terms to approximate sin(x²) dx. Use the first 3 term in the expansion in part (ii) to approximate sin(x²) dx. Round your answer to 6 decimal places.
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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![**Power Series Expansion of the Sine Function**
### Problem 20:
Recall the power series expansion of \(\sin(x)\):
\[
\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
\]
#### i. Deriving the Power Series Expansion for \(\sin(x^2)\)
Use the given expansion to derive a power series expansion for \(\sin(x^2)\).
Show at least 4 non-zero terms.
\[\sin(x^2) = \]
#### ii. Deriving the Power Series Expansion for \(\int \sin(x^2) \, dx\)
Use the given expansion to derive a power series expansion for \(\int \sin(x^2) \, dx\).
Show at least 4 non-zero terms.
\[\int \sin(x^2) \, dx = \]
#### iii. Error Estimation Using Alternating Series Estimation Theorem
Use the Alternating Series Estimation Theorem to determine an upper bound for the error in using only the first 2 terms to approximate:
\[
\int_{0}^{1} \sin(x^2) \, dx
\]
#### iv. Approximation Using the First 3 Terms
Use the first 3 terms in the expansion in part (ii) to approximate:
\[
\int_{0}^{1} \sin(x^2) \, dx
\]
Round your answer to 6 decimal places.
\[
\int_{0}^{1} \sin(x^2) \, dx \approx \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fabc15-e5db-480c-94d9-e42b67b656a8%2F6751e4ec-6aed-4474-a337-70a85ef954d1%2F56ap1ge_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Power Series Expansion of the Sine Function**
### Problem 20:
Recall the power series expansion of \(\sin(x)\):
\[
\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
\]
#### i. Deriving the Power Series Expansion for \(\sin(x^2)\)
Use the given expansion to derive a power series expansion for \(\sin(x^2)\).
Show at least 4 non-zero terms.
\[\sin(x^2) = \]
#### ii. Deriving the Power Series Expansion for \(\int \sin(x^2) \, dx\)
Use the given expansion to derive a power series expansion for \(\int \sin(x^2) \, dx\).
Show at least 4 non-zero terms.
\[\int \sin(x^2) \, dx = \]
#### iii. Error Estimation Using Alternating Series Estimation Theorem
Use the Alternating Series Estimation Theorem to determine an upper bound for the error in using only the first 2 terms to approximate:
\[
\int_{0}^{1} \sin(x^2) \, dx
\]
#### iv. Approximation Using the First 3 Terms
Use the first 3 terms in the expansion in part (ii) to approximate:
\[
\int_{0}^{1} \sin(x^2) \, dx
\]
Round your answer to 6 decimal places.
\[
\int_{0}^{1} \sin(x^2) \, dx \approx \]
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