Provide examples of sequences (an) and (bn) such that an rightarrow + infinity a

Question

Provide examples of sequences (an) and (bn) such that an rightarrow + infinity and bn rightarrow 0 for each of the following (anbn)n N diverges to +infinity. (anbn)n N diverges to +infinity (anbn)n N is unbounded nut does not diverge to +infinity or to -infinity. (anbn)n N converges to some given real number gamma. (anbn)n N bounder but not convergent.

Explanation / Answer

i) an = n bn = 1/sqrt(n) an -> infinity bn -> 0 The product = n *1/sqrt(n) = sqrt(n) diverges to + infinity ii) an = n bn = -1/sqrt(n) an -> infinity bn -> 0 anbn = n*-1/sqrt(n) = -sqrt(n) diverges to - infinity iii) an = n bn = (-1)^n/sqrt(n) an -> infinity bn -> 0 anbn = n * (-1)^n/sqrt(n) = (-1)^n * sqrt(n) has odd subsequences tending to -infinity and even subsequences tending to infinity. Thus, it does not diverge to +infinity or -infinity iv) Let an = n and bn = 1/n an -> infinity bn -> 0 anbn = n * 1/n = 1 for all n Thus, its limit is 1. v) an = n and bn = (-1)^n/n an -> infinity bn -> 0 anbn = n * (-1)^n/n = (-1)^n This is bounded by -1 and 1 (in fact, it only has values -1 and 1, -1 for n odd and 1 for n even), but it is not convergent.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.