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) ²sin(n²)n(2 + cos n)2nn!

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Answered: 4. 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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Determine if the following sequence converges or diverges and find the limit if possible. Sin(n + 1)2(n + 2) {(n – 1)n²(n+ 1) S. n=6
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3. Show that II (1+22) is absolute convergent if and only if |z| < 1. Show k=0 that the product converges to 4. Show that 1+sin( is absolute and uniform convergent for all 2. k=0 5. Show that Res(T(2);-n) = (-1)" n!
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Evaluate the following proposed proof that 1. According to Example 2.1.4, √n² +1 lim nxn+ (cos n)² = 0. 2. Since {1/(n + 1)} is a tail of the sequence {1/n}1, Proposition 2.1.15 implies that 1 n+1 Step 4 is faulty. Step 5 is faulty. Step 2 is faulty. The proof is valid. Step 3 is faulty. Step 1 is faulty. = 1 lim = 0. n→∞ n 1 lim n→∞ n + 1 3. Since 0≤ (cos n)² ≤ 1, the order properties of the real numbers imply that n≤n+ (cos n)² ≤ n + 1, and = = 0. 1 < n+ (cos n)² n 4. The preceding three steps, together with Lemma 2.2.1 (the squeeze lemma), imply that 1 lim n→∞ n + (cos n)² 5. According to Proposition 2.2.5, the limit of a product is the product of the limits, so 0. √n² +1 lim nx n + (cos n)² Which of the following statements best describes this proposed proof? = 0. lim √n² + 1 = 0.

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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!