3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)
3. So far in the course, we have only talked about sequences which converge to some real number. In this problem, we will use the following definition: A sequence (of real numbers) {n} is said to diverge to (positive) infinity if for all KER, there exists some MEN such that for all n ≥ M, xn > K. In this case, we abuse notation and write lim n = +∞ n→∞ (a) Write down a corresponding definition for a sequence {n} which diverges to negative infinity, and use it to show that lim -n³ = -∞ n→∞ (b) Suppose {n} is a sequence satisfying xn> 0 for all n € N, and furthermore 1 lim = 0 n→∞ Xn Show that {n} diverges to positive infinity. (c) Show that a sequence {x} is unbounded above (i.e. the set {xn: :n € N} is unbounded above) if and only if {n} has a subsequence {n} which diverges to positive infinity. (Hint: When trying to find a subsequence {n} which diverges to positive infinity, try proving that every p-tail of {n} is also unbounded. Then, construct the sequence inductively in order to make sure nk+1 > nk.)
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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