Problem 1 - Calculus Early Transcendentals

Question

Give an example of a nonincreasing sequence with a limit.

Step-by-Step Solution

Answer

Question: Give an example of a nonincreasing sequence that converges to a limit and determine its limit. Answer: An example of a nonincreasing sequence that converges to a limit is the sequence (a_n) = 1/n for n ≥ 1. The sequence looks like: 1/1, 1/2, 1/3, .... The limit of this sequence is 0 as n approaches infinity.

Step 1: Example of a nonincreasing sequence

Let's consider the sequence \((a_n)\) defined by \(a_n = \frac{1}{n}\) for \(n \geq 1\). This sequence is nonincreasing since each term is smaller than or equal to the previous term, as \(n\) increases. The sequence will look like this: \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots\)

Step 2: Determine the limit

To find the limit, we need to examine what happens to the sequence as \(n\) approaches infinity. We can do this by finding \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n}\).

Step 3: Use properties of limits

We know that \(\lim_{n \to \infty} \frac{1}{n} = 0\). This is because as \(n\) increases without bound, the fraction \(\frac{1}{n}\) gets smaller and smaller, approaching zero.

Step 4: Conclusion

We found that the nonincreasing sequence \((a_n) = \frac{1}{n}\) has a limit, which is 0. Therefore, the example of a nonincreasing sequence with a limit is: \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots\) with limit 0.