Problem 33 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-2 n}{1+2 n} $$

Step-by-Step Solution

Answer

The sequence converges to -1.

Step 1: Identify the Formula of the Sequence

The sequence given is \(a_n = \frac{1-2n}{1+2n}\). We need to determine whether this sequence converges or diverges and find its limit if it converges.

Step 2: Determine the Behavior as n Approaches Infinity

To analyze the limit of \(a_n\) as \(n\) approaches infinity, we focus on the leading terms in the numerator and the denominator: - Numerator: \( -2n \)- Denominator: \( 2n \)This gives the expression \( \frac{-2n}{2n} = -1 \) as \(n\) approaches infinity.

Step 3: Simplify the Expression

Re-examine the original expression: \[a_n = \frac{1-2n}{1+2n}\\]and divide the numerator and the denominator by \(n\) to simplify:\[a_n = \frac{\frac{1}{n} - 2}{\frac{1}{n} + 2}\].

Step 4: Conclude the Limit

As \(n\) approaches infinity, the terms \(\frac{1}{n}\) in the simplified expression approach 0. Thus, \[a_n \to \frac{0 - 2}{0 + 2} = -1\].Therefore, the sequence \(a_n\) converges to \(-1\).

Key Concepts

Limits of Sequences

In mathematics, a sequence is essentially an ordered list of numbers. As we move down this list, particularly when thinking of sequences, one of the central questions is about the behavior of these numbers "at infinity." This is where the concept of the "limit" comes into play for sequences. The limit of a sequence \{a_n\} as \(n\) approaches infinity is essentially asking what value the numbers in the sequence get closer and closer to as you look further along the sequence. Here's a helpful way to understand it:

• If the numbers in the sequence get closer and closer to a specific number \(L\) as \(n\) becomes very large, we say that \(L\) is the limit of the sequence.
• This concept is crucial because it helps us determine long-term behavior without examining every element in the sequence individually.

For example, in our original sequence, after simplification, we found that the limit as \(n\) approaches infinity is \(-1\). This means as the numbers keep growing, our sequence's values get ever closer to \(-1\). This clear endpoint allows us to say that the sequence converges.

Convergent and Divergent Sequences

Sequences are categorized based on their behavior at infinity into two primary types: convergent and divergent. Understanding these two is key to identifying the nature of sequences.

• Convergent Sequence: A sequence is called convergent if its limit exists; that is, as \(n\) progresses, the sequence approaches a particular number \(L\). It's like zooming in with a camera until the focus lands firmly on \(L\).
• Divergent Sequence: Conversely, a sequence is divergent if it does not approach any specific number. This could mean it heads off to infinity, bounces around without settling, or has no discernible pattern.

Returning to our specific sequence of \(a_n = \frac{1 - 2n}{1 + 2n}\), we say this sequence is convergent because we found its limit to be \(-1\). The sequence stabilizes to this particular value, fitting well within the definition of convergence.

Infinite Limits

When discussing sequences, you might occasionally encounter the term "infinite limits." This occurs in the case of divergent sequences where, instead of approaching a finite number, the sequence's terms grow unbounded either positively or negatively. Imagine a sequence that keeps increasing in size without stopping or one that plummets deeper into the negative without end. These sequences would have infinite limits, essentially suggesting that the sequence doesn't settle on a traditional finite number.

• Positive Infinite Limits: In some sequences, values grow so large they tend toward an infinitely positive value. An example would be the sequence \(a_n = n\) as \(n\to \infty\).
• Negative Infinite Limits: Meanwhile, sequences might trend toward a never-ending negative, like \(a_n = -n\) as \(n\to \infty\).

For the sequence \(a_n = \frac{1 - 2n}{1+2n}\), however, we see it heading towards a fixed number, not an infinite one. This crucial distinction helps solidify our understanding when distinguishing between different sequence behaviors.