Question

State whether the sequence, whose nth terms are indicated, is bounded and whether it is eventually monotone, increasing, or decreasing. (Select all that apply.) n-1/n, n 3 bounded O not bounded monotone not monotone increasing decreasing

Answer 1

The sequence (n - 1)/n is bounded and eventually monotone increasing, as it approaches 1 but remains less than it for all n, and each term grows closer to 1 as n increases.

Step-by-step explanation:

The sequence in question is given by the nth term (n - 1)/n. To determine if this sequence is bounded and whether it is monotone (increasing or decreasing), we need to analyze its behavior as n increases.

Firstly, consider the sequence bounds. As n goes to infinity, the term (n - 1)/n approaches 1, but it will always be slightly less than 1 since we are subtracting 1 from the numerator. Therefore, the sequence has an upper bound of 1. Because all terms are positive (for n > 0), we can also note that this sequence has a lower bound of 0. Hence, the sequence is bounded.

For monotonicity, we observe the sequence as n increases. If we take two consecutive terms when n is large enough, say (n - 1)/n and (n)/n+1, we see that the second term is larger because the difference between the numerator and the denominator is getting smaller. This implies that each subsequent term is closer to 1 than the previous. Thus, the sequence is eventually monotone increasing.