Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{5-\frac{1}{n}\right\\}$$

Short answer

Converges, limit is 5.

Step-by-step solution

Identify the sequence terms as n approaches infinity

The given sequence is \( \{a_n\} = \{5 - \frac{1}{n}\} \). The term \( a_n = 5 - \frac{1}{n} \) depends on the variable \( n \). As \( n \) increases (approaches infinity), we evaluate the behavior of the sequence.

Analyze the behavior of the fraction \( \frac{1}{n} \) as \( n \) approaches infinity

The term \( \frac{1}{n} \) becomes very small as \( n \) approaches infinity because the denominator grows without bound. Specifically, \( \frac{1}{n} \to 0 \) as \( n \to \infty \).

Determine the limit of the sequence

Substituting the behavior of \( \frac{1}{n} \) into the sequence term, we have \( a_n = 5 - \frac{1}{n} \). As \( \frac{1}{n} \to 0 \), \( a_n \to 5 - 0 = 5 \). Therefore, the limit of the sequence is 5.

Conclude whether the sequence converges or diverges

Since the sequence \( \{a_n\} = \{5 - \frac{1}{n}\} \) approaches a fixed limit of 5 as \( n \to \infty \), the sequence is convergent.

Key concepts

Limit of a Sequence

A sequence is a list of numbers in a specific order and understanding its limit can help determine the sequence's behavior as it progresses. For a sequence given by \( \{a_n\} = \{5 - \frac{1}{n}\} \), the term \( a_n \) is expressed as a function of \( n \). The limit of a sequence is what the terms of the sequence approach as \( n \) becomes very large, or more technically, as \( n \) approaches infinity.

In our example sequence, as \( n \) increases, \( \frac{1}{n} \) decreases because larger denominators make the fraction smaller. So, \( \frac{1}{n} \) approaches 0. Consequently, \( a_n = 5 - \frac{1}{n} \) approaches \( 5 - 0 = 5 \). This number, 5, is called the limit of the sequence because it's the value the terms of the sequence get closer to as \( n \) becomes very large.

When a sequence has a specific limit, we say that it converges to that limit. Thus, the sequence \( \{a_n\} = \{5 - \frac{1}{n}\} \) converges to the number 5.

Behavior of Sequences at Infinity

As sequences extend towards infinity, their behavior gives us clues about their convergence or divergence. In simple terms, we look for patterns or trends in the values of the sequence as \( n \) gets very large.

In the sequence \( \{5 - \frac{1}{n}\} \), notice how \( \frac{1}{n} \) becomes smaller the larger \( n \) gets. At the start, the terms might differ significantly from the limit, but gradually, as \( n \) becomes enormous, the terms become practically indistinguishable from the limit; in this case, closer and closer to 5.

It's important to recognize that a sequence approaches its limit; it never *necessarily* reaches it, especially when we're talking about infinity. But for practical purposes, after numerous terms, the difference between the terms of the sequence and the limit becomes tiny.

Infinity in Sequences

Infinity plays a crucial role in understanding sequences and their limits. When we say that \( n \) approaches infinity, we refer to letting \( n \) grow indefinitely larger. This concept helps us make sense of sequences that don't end and always provide new terms for us to consider.

In the sequence \( \{5 - \frac{1}{n}\} \), as \( n \) heads towards infinity, \( \frac{1}{n} \) heads closer to zero. This is an example of how infinity helps us deduce the long-term behavior of a sequence. Essentially, infinity is a way to imagine an infinite number of steps or terms in a sequence to observe patterns or eventual outcomes.

Understanding how sequences behave at infinity is essential because not all sequences have limits. Some sequences diverge, meaning they do not settle on a particular value but hang around specific trends. However, knowing that a sequence approaches a limit gives us powerful insights into its nature and properties.