Q1-A) Test the following sequence whether it is converging or diverging:-a. - [8m - 25/m] sin( 1 ) + ceath [msin(ansin()+coshsim(!]MT

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Answered: Q1-A) Test the following sequence…

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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The sequence (xm) = (1, 3, 1, 3, 1, . .., 1, 3, ...) is: O Converge to 3. O Divergent. Converge to 0 and 3. Converge to 0. O Converge to 1.
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.
Mark each sequence {n} having the property that lim infan> inf{n: n € N}. n+∞0 U U 0 U Xn = = {cos(nπ/3)} (-1)"n n+1 Xn = In = 1 Xn= n 1 when n is odd, and n = 1+ - when n is even. n xn = (-1)" n n- (-1)"
Needed to be solved correclty in 10 minutes and get the thumbs up please show neat and clean work
Evercise 5. Consider the sequence (fn)n defined by fn(x) = #na for r >0. 2. Show that for a > 0, (fn)n converges uniformly to f on [a, o).
Let n E N* and a, BE R. O a. The sequence a + is increasing %%3D O b. The sequence (-1)" -is decreasing The sequence p - is decreasing (4/3)" a 1. The sequence p is neither increasing nor decreasing (x4)"a e. The sequence a is increasing 彩 Type here to search AAc
cosh(2r) – 1 6. (a) Show that sinh?(z) = 2 (b) Find the nth term of the following sequence {1,8, 10, 26, 46, · - - } for all n > 1.
n=D10 1 Generate 11 polynomials (for n=0, n=1,..., n=10) from the sequence P,(x) = d" -(x² –1)" %3D 2" n! dx" n=0 You may use technology or write a computer program to derive the sequence.
111. Newton's method The following sequences come from the recursion formula for Newton's method, f(x,) Xn+1 = Xp f'(x„)" Do the sequences converge? If so, to what value? In each case, begin by identifying the function ƒ that generates the sequence. Xл a. xo = 1, Xn+1 = Xn 2.x,, tan x, b. Хо — 1, х,+1 — Хд sec? xn c. Xo = 1, Xp+1 = Xn - 1
Let 0 < 0 < 1 be a rational number. Show that the sequence {cos(nT0)}-1 is divergent.

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Q1-A) Test the following sequence whether it is converging or diverging:- 2n 2. = [0.25m 2 ) ] sin()+cosh n sin( h[n sin( an s sin() 27