sFind the sequence yt that satisfies the differential equation.mbrol of at: besy₁₁ - 2y += ₁ + ty/=2=15"and the initial conditions foYo= 6 and y₁ = 6tas Micallery• Describe the behaviour of this sequenceas t increases...

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Answered: s Find the sequence yt that satisfies…

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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Find the limit of the sequence. Hint: Remember that cos n always satisfies -1 ≤ cos n ≤ 1. an = (cos n) 7n
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z„} e So for n > N. Then: (i) (1– p)zn N; (ii) {zn¢n} is increasing for all n > N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- ponding sequence {zn} E So. Then z, is positive, Zn 2 yn, and Yn = Zn – Pnyo(n) 2 (1– p)zn, n>N> no, so (i) is proved. It easy to see that z, E So implies lim b,(Az,)“ = 0; n00 otherwise we would eventually have Az, > 0 contradicting z, E So. Similarly, lim a„A(b,(Azn)“) = 0. A summation of equation (1.1) then yields 9szs+1 Zn+1 LIs. s=n s=n s=n Summing once more, we obtain as t=s s=n or Azn 2-Zn+1Qn. Hence, A(zn&n) = PnAzn + Zn+1Ao, > Zn+1(A0n – OnQn) =0 since {0,} is a solution of the difference equation (Ao, – Qnºn) = 0. Therefore, {z,9n} is increasing and this completes the proof of the lemma.
= 4n – 4. Let an be the sequence defined for N > 0 by an Find the first 5 terms a0, a1, ..., a4. Let Sn E-o ai. 0. Find the first 5 terms of the sequence Sn, starting with n = Let tn E, si. i=0 = 0. Find the first 5 terms of the sequence tn, starting with n =
11. Find 4he sum oť the Inffinite geom. 2. series 2-1+ Ya -4
Let a, = 2" cos(n7) for n21. Write out the first 8 terms of the sequence (a, through ag). Determine lim a, . Note: You must show all your work and justify your steps carefully. а. b. Determine if Ea, converges (absolutely or conditionally) or diverges. Note: You must show с. all your work and justify your steps carefully.

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s Find the sequence yt that satisfies the differenti al equation. mbrol of at: bes y₁. - 2y + ₁ + 7y/₂ = 15 and the initial conditions fo Yo = 6 and y₁ = 6 tool collarin • Describe the behaviour of this sequence as t increases... the del siti di bro