Question

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

Short answer

The sequence converges to 2.

Step-by-step solution

Write the sequence formula

The given sequence is \( \{ a_n \} = \left\{ \frac{2n}{n+1} \right\} \). We need to determine whether this sequence converges or diverges.

Analyze the limit of the sequence as n approaches infinity

To determine convergence, find \( \lim_{{n \to \infty}} \frac{2n}{n+1} \). For large \( n \), the degree of \( 2n \) in the numerator is the same as the degree of \( n+1 \) in the denominator. This suggests that the limit is the ratio of the leading coefficients of these polynomials.

Simplify the sequence expression

Rewrite the sequence expression as \( \frac{2n}{n+1} = \frac{2n}{n} \cdot \frac{1}{1+1/n} = 2 \cdot \frac{1}{1+1/n} \).

Determine the limit

Since \( \lim_{{n \to \infty}} \frac{1}{n} = 0 \), it follows that \( \lim_{{n \to \infty}} \frac{1}{1+1/n} = 1 \). Thus, \( \lim_{{n \to \infty}} \frac{2n}{n+1} = 2 \cdot 1 = 2 \).

Conclude about convergence

Because the limit of the sequence exists and is a finite number, the sequence \( \{ a_n \} \) converges, and the limit is 2.

Key concepts

Understanding Limits

In mathematics, a limit helps us understand the behavior of a function or sequence as its inputs approach a certain value. For sequences, this means identifying what number the sequence gets closer to as it progresses towards infinity.
If a sequence gets closer and closer to a specific number, we say the sequence converges, and that number is the limit. If a sequence doesn't seem to settle down to any number, we say it diverges.
When analyzing the limit of a sequence like \( \left\{ \frac{2n}{n+1} \right\} \), it's crucial to consider what happens as \( n \) becomes very large. This analysis typically involves identifying dominant terms and their coefficients, which leads us to another essential concept—the ratio of leading coefficients.

Infinite Sequences

Infinite sequences are ordered lists of numbers that continue indefinitely. Each number in this list is called a term, and it's usually denoted by \( a_n \), where \( n \) indicates the position of the term in the list.
The sequence \( \left\{ \frac{2n}{n+1} \right\} \) is an example of an infinite sequence. It's expressed as a formula that describes how each term in the sequence is calculated based on its position \( n \).
Infinite sequences can either converge, meaning they approach a specific limit as \( n \) gets larger, or diverge, meaning they do not settle on any particular value. Understanding infinite sequences helps us make sense of various mathematical patterns and behaviors.

Ratio of Leading Coefficients

In the context of polynomials, the leading coefficient is the coefficient of the term with the highest degree. When dealing with sequences or functions that involve rational expressions like \( \frac{2n}{n+1} \), the ratio of the leading coefficients becomes crucial.
For the given sequence, the numerator has a leading coefficient of 2 (from \( 2n \)), and the denominator has a leading coefficient of 1 (from \( n \)).
When \( n \) approaches infinity, terms without \( n \) become negligible. Thus, the sequence’s behavior is governed mainly by these leading coefficients. This simplifies finding the limit of the sequence: \( \lim_{{n \to \infty}} \frac{2n}{n+1} = \frac{2}{1} = 2 \).
Knowing this, we see that the ratio of leading coefficients is a powerful tool in determining sequence convergence and limits.