1. Determine whether the following sequence converges or diverges.(a) {3n² + 10) (b) {1.3.5 (22)-1)} (c) (sin(n+1)]1.3.5....2.4.6....en1)²

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Answered: Determine whether the following…

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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4. Determine whether the following sequences converge. Rigorously justify your answers. You may use any theorem or example proved during the lecture course. (a) an = (b) bn (c) cn = n² +3 (n+3) ² sin(n²) n(2 + cos n) 2n n!
Given the series s (-1)"(In(n))² n2 (a) Using the Alternating Series Error bound, determine the smallest value of N for which |s- sy < .01. (NOTE: if solve does not work, try nsolve. If that does not work, print the first several terms of the sequence |a,| and observe your value of N). (b) Use the nsum command to find Sy for your value of N in part a) and the actual sum s. (In(n))" n? (c) We now want to test for absolute convergence, namely, does con- verge. Apply the Integral Test on an appropriate function to show this series does converge. (d) Using the Integral Test Error bound, determine the smallest value of N for which |8 – SN| < .01. (Refer to the first part of the hint for part (a) here as well).
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
2. Determine the limit of the following sequences or state if the sequence diverges. Use the technique in square brackets. TL (i) Igeometric sequence] an = „()- (ii) a, = 11 + [limit laws] 15 n2 12n2 - n+1 n2 – 6 (iii) an [limit laws]
1. Determine if each sequence converges or diverges. Find the limit of any convergent sequences. 4+n-3n² 4n²+1 (a) { (b) {ln(12n+2) - In(9+4n)} (2n-1)! n! (c) {. (d) cos(nπ) 2} n (e) {(1 + ²)"} (f) {n(√n² +1 − n)} 82n n!
(a) Write the first 5 terms of the sequences with general terms an below and indicate wnether they are convergent or divergent. 2n + 1 1 (iv) (-1)" (2) an = (ii) an = (2n – 1).3" (iii) an = 2 + An = 1+ n3 1- 3n %3D
1 +nx The sequence fn(x) = is 72 Select one: Oa. diverges Ob. converges uniformly but not pointwise c. converges pointwise but not uniformly

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Determine whether the following sequence converges or diverges. (a) { 3n² + 10} (b) D} (c) 1.3.5.... (2n-1) 2.4.6.... (2n) en [sin²(n+1) (n + 1)²