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(-1)"x2" (2n)! +0o Σ For all x E R, cos x = n=0 a. Find a power series that is equal to x cos(x²) for all x e R. b. Differentiate the series in item 1(a) to find a power series that is equal to cos(r²) – 2x² sin(x²) for all x E R. +0o c. Use the result in item 1(b) to prove that n=0 (-16)"(4n+1) (2n)! cos(4) – 8 sin(4).

(-1)"x2" (2n)! +0o Σ For all x E R, cos x = n=0 a. Find a power series that is equal to x cos(x²) for all x e R. b. Differentiate the series in item 1(a) to find a power series that is equal to cos(r²) – 2x² sin(x²) for all x E R. +0o c. Use the result in item 1(b) to prove that n=0 (-16)"(4n+1) (2n)! cos(4) – 8 sin(4).

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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For item 1 po use theorems under Functions as Power series
ఆ (-1)"xW 1. For all x ER, cos x = Σ (2n)! n=0 a. Find a power series that is equal to x cos(x) for all x E R. b. Differentiate the series in item 1(a) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. +00 (-16)"(4n+1) (2n)! c. Use the result in item 1(b) to prove that ) = cos(4) - 8 sin(4). n=0
Transcribed Image Text:ఆ (-1)"xW 1. For all x ER, cos x = Σ (2n)! n=0 a. Find a power series that is equal to x cos(x) for all x E R. b. Differentiate the series in item 1(a) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. +00 (-16)"(4n+1) (2n)! c. Use the result in item 1(b) to prove that ) = cos(4) - 8 sin(4). n=0
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