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Siven the following: 00 5(-1)"x2n '(-1)"x²n cos(x) = for all x E R (2n)! n=0 ) Find a power series that is equal to x cos(x²) for all x ER ) Afterwards, use differentiation on the series found in "a)" to find a power series that is equ o cos(x³) - 2x² sin(x²) for all x € R ) Finally, use the answer found in "b)" to prove that the following statement is valid: cos(4) – 8 sin(4) = 5-16)"(4n + 1) (2n)! for all x E R n=0

Siven the following: 00 5(-1)"x2n '(-1)"x²n cos(x) = for all x E R (2n)! n=0 ) Find a power series that is equal to x cos(x²) for all x ER ) Afterwards, use differentiation on the series found in "a)" to find a power series that is equ o cos(x³) - 2x² sin(x²) for all x € R ) Finally, use the answer found in "b)" to prove that the following statement is valid: cos(4) – 8 sin(4) = 5-16)"(4n + 1) (2n)! for all x E R n=0

Advanced Engineering Mathematics
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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Given the following: 5-1)"x2n (2n)! (-1)"x²n cos(x) for all x E R n=0 a) Find a power series that is equal to x cos(x2) for all x E R b) Afterwards, use differentiation on the series found in "a)" to find a power series that is equal to cos(x2) - 2x2 sin(x²) for all x ER c) Finally, use the answer found in "b)" to prove that the following statement is valid: cos(4) – 8 sin(4) = S-16)*(4n + 1) (2n)! for all x ER n=0
Transcribed Image Text:Given the following: 5-1)"x2n (2n)! (-1)"x²n cos(x) for all x E R n=0 a) Find a power series that is equal to x cos(x2) for all x E R b) Afterwards, use differentiation on the series found in "a)" to find a power series that is equal to cos(x2) - 2x2 sin(x²) for all x ER c) Finally, use the answer found in "b)" to prove that the following statement is valid: cos(4) – 8 sin(4) = S-16)*(4n + 1) (2n)! for all x ER n=0
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