Question

Give an example of a nonincreasing sequence with a limit.

Short answer

Question: Provide an example of a nonincreasing sequence with a limit. Answer: The sequence \(a_n = \frac{1}{n}\) is a nonincreasing sequence with a limit of 0 as n approaches infinity.

Step-by-step solution

Identify a nonincreasing sequence

A simple example of a nonincreasing sequence is the sequence of numbers that approaches zero. A good example of this sequence is using reciprocals of natural numbers (i.e., \( \frac{1}{n} \)) . As n increases, the reciprocal value decreases. So, the sequence can be defined as \(a_n = \frac{1}{n}\).

Confirm that the sequence is nonincreasing

To confirm that the sequence is nonincreasing, we need to show that for every pair of consecutive terms a_n and a_(n+1), a_(n+1) is either equal to or less than a_n. Since \(a_n = \frac{1}{n}\) and \(a_{n+1} = \frac{1}{n+1}\), we need to show that: \( \frac{1}{n+1} \leq \frac{1}{n} \). As both the numerators are equal, and \(n+1 > n\), it is enough to conclude that \( \frac{1}{n+1} \leq \frac{1}{n}\), confirming that the sequence is nonincreasing.

Determine the limit of the sequence

Now, we need to find the limit of the sequence as n approaches infinity. The limit is represented as: \( \lim_{n \to \infty} \frac{1}{n} \). Since the denominator is increasing indefinitely while the numerator remains constant, the value of the fraction approaches zero: \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Conclusion

An example of a nonincreasing sequence with a limit is \(a_n = \frac{1}{n}\), which approaches a limit of 0 as n approaches infinity.