Given that m is a positive integer and {D„} be the sequence with D = m and (Dn-1 + Dn-1 m D, = ,n > 1. Find lim Dn n-+00
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- Suppose that the sn satisfies both limn→∞ s2n = 2 and limn→∞ s2n+1 = 2. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 2.) Show that also limn→∞ sn = 2.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.Consider the sequences (an) and (bn) given by an= (−1)nn2, and bn= 2n. Decide if the sequence (bn/an: n≥1) converges. If it does, what is its limit? What about (an/bn: n≥1)? Does it converge too?
- 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.Find the limit of the following sequence or determine that the sequence diverges. {(sin 6n)/(5n)}
- Attached. I have to find four terms of the Laurent series.Determine if this sequence converges og diverges: n3/(cos n+ 5n)Suppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using the definition of limit, explain why if limn → ∞ sn = s, then also limn → ∞ tn = s.