3. Prove from first principles that the following sequences converge. (Hint: your first jobis to figure out, informally, what the limit is in each case.)an =Cn =n+1n+ 2'{n odd400/n-1/(400m²) n even"0bn=dn =5n4n²-3'{ "/(1+7n/(1+n²) 1≤n ≤ 900n> 901

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Answered: 3. Prove from first principles that the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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22. Suppose that (un) is an increasing sequence and (u2m) is convergent. Show that (un) is convergent. 23. Let 0 € (0, T/2), 2" tan 2n Show that (un) and (vn) are adjacent. Find their limits. Un = 2" sin 2n and Un
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4. (a) Is each of the following a Cauchy sequence in R? Justify your answer. i. The real sequence {xn} defined by xn = 3ne Vn5+3n2+2' Jn E N. ii. The real sequence {yn} defined by yn = n(-1)" Vn E N. (b) Let {z,} be a real sequence defined by 1 z1 = 5, zn+1 = V5zn + 12 - for n > 1. n Prove or disprove that {zn} is a convergent sequence.
3. Let an = (3n +4n)¹/n (a) Prove that for all n € N, 4 < an = 4(1 + (²)n)¹/n < 4.2¹/n (b) Prove that the sequence {an}=1 is convergent and find its limit.
3. For each of the sequences below, determine whether it converges, and if it converges, guess its limit and prove your guess using the e - N approach. (b) {} (0) {2"} {} {} (m) {} {} n²+3 (d) {73 + (1)"} (e) {(-1)"n} 2n+1 (f) 7n3+8n (g) 9n2-18 6n+18 ,(i). 2n+1+5 2-7 2n-31 4. Prove the following: (a) lim = (b) lim [(-1)"n] = 0 (c) lim =0 ,b€R %3D %3D L3n+2 11+00 (d) lim Vn +n-n= 0 (e) lim Jait = () lim = %3D %3D Vn+1+vn T 00
Prove from first principles that the following sequences converge.
5.2. Consider the sequence defined by xo = a > 0 and a Xk + Xk Xk+1 = Prove that this sequence converges to X* = Vā and that the convergence rate is quadratic, and determine the rate constant.

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3. Prove from first principles that the following sequences converge. (Hint: your first job is to figure out, informally, what the limit is in each case.) an = Cn = n+1 n+ 2' { 400/n -1/(400m²) n odd n even " 0 bn = dn = 5n 4n²-3' { "/(1+ 7 n/(1+n²) 1≤ n ≤ 900 n> 901