1. Suppose that 0 ≤ a, b, for every positive integer n. Which of the following statements is always true?(a) If lim b= 0, then Σa is convergent.11-900(b) Ifa, is divergent, then so is(e) If b, is divergent, then so isnal(c) Ifa, is convergent, then so is bn=1n=1(d) none of these are always true.2. Write f(x)(a)(b)00001+42²4243(-4)2+6n=0(c) Σ 4,2+6n=0DO(-4)2+3(d)(e) Σ(-4)-2bn=1(b) 0(c) None of these.(d)(e) 4Onas a power series centered at 0.3. Find the radius of convergence of the seriesAWI4" (x-3)"n!

**Note: According to the Bartleby guideline I should answer the first one only.Comparison test…

Answered: 1. Suppose that 0 ≤ a, b, for every…

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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If a
1. Let (x) and (yn)₁ be two sequences of real numbers. Assume that there exists a positive integer N with the property xn = yn for all n ≥ N. Show that lim xn exists if and only if lim yn exists.
10. Let {an}, {bn} and {cn} be sequences of real numbers. Which of the following statements are true? I. If {an} is convergent and {bn} is bounded, then {a„bn} is convergent. II. If lim an = L and lim b, = 0, then n-00 lim anbn = 00. n-00 III. If a, #0 for all n e Z+ and lim n→∞ An 0, then %3D lim an = 00. n-00 IV. If lim an = 0 and {bn} is bounded, then n-00 lim anbn = 0. n00 V. If b, < an < Cn for all n e Zt, {bn} and {cn} are convergent, then {an} is also convergent. VI. If {an} is unbounded, then {an} is divergent. (a) I, II (b) I, IV (c) IV, VI (а) , II, V (е) 1, IV, V
#11
1. Which of the following statement (s) is/are TRUE? I: If lim an 0 then (In converges II: If 0 (In COllVerges. III: If an>0 and an (In D) II and III E) 1. II and II A) Only I B) Only II C) Only III
Suppose that we wanted to show that lim does not exist. X - -1 X+1 Decide whether the argument below is a true or false method of proving this. -1 + (-1+)= We could take the sequence 1 Since -1+ 1 (sum of convergent sequences) but -n +1• we could prove that 1 (-n+1) is unbounded (and therefore divergent) and use this to conclude that lim X --1 does not exist (DC for limits). X +1 O True False
n2
number 1
Sn, 4. Let Sn cos(2). Use the definition to find lim supsn and lim inf 3 then explain why sn has no limit. (hint Thm 10.7) =

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