For item 1 po use theorems under Functions as Power series ఆ (-1)"xW1. For all x ER, cos x =Σ(2n)!n=0a. 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) forall 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

Answered: Image /qna-images/answer/b5226c63-b515-4380-9e84-03546b2f87ff.jpg

(-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).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 50E

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