Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{\frac{n-1}{n}-\frac{n}{n-1}\right\\}, n \geq 2$$

Short answer

The sequence converges and its limit is 0.

Step-by-step solution

Simplify each term in the sequence

Consider the given sequence \( a_n = \frac{n-1}{n} - \frac{n}{n-1} \). Simplify each fraction individually:1. Simplify \( \frac{n-1}{n} \) to get \( 1 - \frac{1}{n} \).2. Simplify \( \frac{n}{n-1} \) using long division or by approximating for large \( n \), which is approximately \( 1 + \frac{1}{n-1} \) for large \( n \).

Combine the simplified terms

Substitute the simplified forms back into the sequence expression:\[ a_n = \left( 1 - \frac{1}{n} \right) - \left( 1 + \frac{1}{n-1} \right) \].Combine the terms:\[ a_n = 1 - \frac{1}{n} - 1 - \frac{1}{n-1} = -\frac{1}{n} - \frac{1}{n-1} \].

Analyze the behavior as \( n \to \infty \)

As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{1}{n-1} \) both tend towards zero. Therefore, the sum of these terms, \( a_n = -\frac{1}{n} - \frac{1}{n-1} \), also tends towards zero.

Conclusion about convergence

Since the terms of the sequence approach zero in the limit, the sequence \( \{ a_n \} \) converges to \( 0 \) as \( n \to \infty \). Therefore, the limit of the sequence is 0.

Key concepts

Limit of a sequence

A sequence is a list of numbers ordered in a specific way, and the limit of a sequence helps us understand the long-term behavior of its terms. In other words, it describes what happens to the terms of the sequence as we progress further and further into the list.

When we talk about the limit of a sequence, we're essentially asking if the terms of the sequence are getting closer and closer to a particular number as the sequence continues. If this happens, the sequence is said to converge, and that particular number is called the limit.

For example, in the sequence we are looking at, we found that as the index of the sequence, denoted by \( n \), becomes very large, the sequence \( a_n = -\frac{1}{n} - \frac{1}{n-1} \) tends towards zero. This means that the sequence converges, and the limit of the sequence is 0.

Simplifying fractions

Simplifying fractions is a crucial step when analyzing sequences, as it makes the expressions easier to work with. To simplify a fraction, we try to express it in its most reduced form. This can involve looking for common factors in the numerator and denominator or approximating for large values.

In our exercise, the sequence given is \( \frac{n-1}{n} - \frac{n}{n-1} \). These shows up as two separate fractions. To simplify them:

• \( \frac{n-1}{n} \) simplifies to \( 1 - \frac{1}{n} \). This is because dividing each term in the numerator by \( n \) separates it into terms that are easier to handle.
• \( \frac{n}{n-1} \) is approximately \( 1 + \frac{1}{n-1} \), particularly as \( n \) gets large. This simplification is done because \( n/(n-1) \) can be split into parts that show both the principal "1" and the small difference \( \frac{1}{n-1} \) that remains.

This way, our sequence becomes easier to analyze, making it clear which terms will dominate as \( n \) increases.

Behavior as n approaches infinity

The behavior of a sequence as \( n \) approaches infinity is an essential concept in understanding whether it converges and, if so, to what value. As \( n \) becomes very large, we look at what happens to the expression terms in the sequence.

For the sequence \( a_n = -\frac{1}{n} - \frac{1}{n-1} \), as \( n \) increases, both \( \frac{1}{n} \) and \( \frac{1}{n-1} \) tend towards zero because dividing one by an increasingly larger number results in a smaller quotient.

This is a classic example of analyzing limits in sequences. As these fractions tend to zero, the sum \( a_n = -\frac{1}{n} - \frac{1}{n-1} \) also tends to zero. This tendency determines the limit of \( a_n \) and confirms that the sequence converges to a limit of 0. Understanding this behavior helps us conclude the sequence's behavior at infinity and provides a clearer picture of its overall pattern.