(-1)"x2n(2n)!+00Σ1. For all x E R, cos x =n=0a. 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(x²) – 2x sin(x²) forall x E R.to° (–16)"(4n +1)(2n)!+o0c. Use the result in item 1(b) to prove thatcos(4) – 8 sin(4).%3Dn=0

Substitute x2 for x in the given series of cosx and simplify to obtain the series of cosx2.…

Answered: 1. For all x e R, cos x = -1)"x²n (2n)!…

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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H.W 2/ Answer the following points Find the first three nonzero terms of Maclaurin series for the functions f(x) = e and other function f(x) = e ii. By depending on point (i), find the series for function f(x) = Cosh(x) where Cosh(x) = te Ans/ ) e=1+++ ti) Cosh(x) = 2 e =1- - +1 Cosh(x) = 1+E
Which of the followings are divergent series? 1. E 5+2" n31 2nt2 II. 1 00 1 4n1 II. E(sin1)" IV. cos(nn)7"/2 n-1 O a. I, II O b.I, IV O d. II, IV O e. II, II
4. It can be shown using material from a later section that Set ² d da is given by (−1)na²n+1 n!2n (2n + 1) n=0 for any positive value of a. When a is positive, this series is an alternating series. (a) Approximate dx to four decimal places accuracy by using the Alternating Series Test and its remainder theorem. (Be careful to not round intermediate steps.) (b) Using the same number of terms as you used in your sum in part (a), compute the midpoint approximation Mn for the integral fe-t³² da (c) Which approximation is more accurate? This should be determined using just parts (a) and (b).
The questions are multiple-choice, all of them with a set of possible answers, which are true or false.
+00 (-1)"2n (2n)! 1. For all x € R, cos a = n=0 (a) Find a power series that is equal to x cos(x2) for all a E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x2) – 2x2 sin(x2) for all a eR. (c) Use the result in (1b) to prove that +oo (-16)" (4n + 1) (2n)! = cos(4) – 8 sin(4). n=0
Given cos(x) = (-1)" n=0 (2n)! -x2n for all x ER, Express the integral cos x dx as a power series.
(c) cOS(7)e " (HINT: write the series as the sum of two series, one of which has only non-negative terms and the second only has negative terms.)
(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. (c) Use the result in (1b) to prove that +oo Σ (-16)"(4n+ 1) (2n)! cos(4) – 8 sin(4). n=0 W!
If f(t)= t2-2t where 0< t

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