Question

Determine whether or not each of the following sequences is (i) bounded, (ii) monotone, and (iii) convergent. Find the limit of any convergent sequence.

a. {1/5+(-1)ⁿ}

Answer 1

The given sequence {1/5+(-1)^n} is bounded but not monotone and does not converge.

Step-by-step explanation:

The given sequence is {1/5+(-1)^n}. To determine whether this sequence is bounded, monotone, and convergent, we need to analyze its behavior as n increases.

The sequence is bounded if there exists a real number M such that |a_n| ≤ M for all n. In this case, the sequence is bounded because |1/5+(-1)^n| ≤ 1/5 + 1 = 6/5 for all n.

A sequence is monotone if it is either increasing or decreasing. In this case, the sequence is not monotone because it alternates between positive and negative terms.

To determine if the sequence is convergent, we need to find its limit. The sequence does not have a limit because it oscillates between two different values, 6/5 and -4/5, as n increases.