' Write out the first five terms of the sequence with, [(+)"] determine whether the sequence converges, and if so find its limit. Enter the following information for a₁ = (n+44) ". a1 a₂ = a3 = a4 = a5 = n +4 n lim = n+1 (Enter DNE if limit Does Not Exist.) Does the sequence converge (Enter "yes" or "no").
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- Determine if the sequence {an}∞n=1 with an=4n^2/√ (4n4+6n2)−converges. If it converges write its limit.Suppose that the sn satisfies both limn→∞s2n = -1 and limn→∞s2n+1 = -1. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to -1.) Show that also limn→∞sn = -1.E. i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge. .
- i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.i. Suppose that the sn satisfies both limn→∞s2n = 3 and limn→∞s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.Determine whether the sequence (3n)/(4n-1) converges or diverges. Show all necessary calculations and clearly state conclusion.
- uppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.Suppose a1=5sin15,a2=25sin125,a3=125sin1125,a4=625sin1625,a5=3125sin13125,...a1=5sin15,a2=25sin125,a3=125sin1125,a4=625sin1625,a5=3125sin13125,... a) Find an explicit formula for anan: . b) Determine whether the sequence is convergent or divergent: .(Enter "convergent" or "divergent" as appropriate.) c) If it converges, find limn→∞an=limn→∞an=Show that the infinite sequence 5. an = √ (n2 + 1) − √ (n2 − 1) (n ≥ 1) converges by showing that it is monotone and bounded. You do not need to find the limit of the sequence
- Let (sn) be a sequence with lim sup sn = 10 and lim inf sn = 5.(a) Show that (sn) is bounded.(b) Construct a sequence (sn) with lim sup sn = 10 and lim inf sn = 5 and withinfinitely many subsequential limits. (It is redundant to say this but just toemphasize: you need to prove that your constructed sequence has infinitelymany subsequential limits and that lim sup sn = 10 and lim inf sn = 5.an = (−1)n 3(sq.root of n) Determine whether the sequence converges or diverges. If it converges, find the limit.Prove that a sequence {an} converges to 0 if and only if the sequence of absolute values 5 | an | 6 converges to 0.