Problem 29 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) 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: Understand the Sequence

We have the sequence \(a_n = \frac{1 - 2n}{1 + 2n}\). Our goal is to determine whether this sequence converges or diverges as \(n\) approaches infinity.

Step 2: Analyze the Behavior of the Sequence

To analyze the behavior as \(n\) goes to infinity, examine the leading terms in the numerator and the denominator. These are \(-2n\) (from \(1 - 2n\)) and \(2n\) (from \(1 + 2n\)).

Step 3: Simplify the Sequence for Large n

Factor out \(n\) from both the numerator and the denominator:\[a_n = \frac{-2n(1 - \frac{1}{2n})}{2n(1 + \frac{1}{2n})}\]. This simplifies to:\[a_n = \frac{-2 (1 - \frac{1}{2n})}{2 (1 + \frac{1}{2n})}\].

Step 4: Take the Limit as n Approaches Infinity

As \(n\) approaches infinity, the terms \(\frac{1}{2n}\) in the expression \(1 - \frac{1}{2n}\) and \(1 + \frac{1}{2n}\) tend to zero. Therefore, the expression simplifies to:\[\lim_{{n \to \infty}} a_n = \frac{-2 \cdot 1}{2 \cdot 1} = \frac{-2}{2} = -1\].

Step 5: Conclusion on the Convergence of the Sequence

Since the limit exists and is finite, \(a_n = \frac{1-2n}{1+2n}\) converges and its limit is \(-1\).

Key Concepts

Limits of Sequences

A sequence is a list of numbers, usually defined by a formula for its general term, like \(a_n = \frac{1-2n}{1+2n}\). A sequence can either converge or diverge. When we talk about the limit of a sequence, we're interested in what value the sequence approaches as the variable \(n\) becomes very large.
If the sequence settles at a specific number, we say it **converges** to that limit. For instance, our sequence has a limit \(-1\), which means no matter how large \(n\) becomes, the values of \(a_n\) get closer and closer to \(-1\).
Key things to remember about limits of sequences:

• If a sequence has a limit, it converges.
• If a sequence keeps increasing or decreasing without bound, it diverges.
• Some sequences might oscillate between values and never settle on a single number, indicating divergence.

To find the limit, we often simplify the expression for large \(n\). This involves keeping the leading terms or factoring to simplify. For instance, our sequence was simplified by factoring out \(n\), focusing only on the dominant terms, which helped us see that the sequence converges to \(-1\).

Infinity in Sequences

Infinity is a powerful concept in sequences, used to describe what happens when numbers grow without bound. When analyzing sequences, infinity helps us understand how the terms behave as \(n\) becomes very large, virtually limitless.
In the context of sequences, you will observe:

• A sequence might approach infinity if its terms grow indefinitely large.
• If terms of a sequence approach a fixed number, the limit is usually a finite value, such as \(-1\) in our example.
• Conversely, if they grow larger positively or negatively without settling, the sequence diverges.

Infinity plays a critical role in determining convergence and divergence. For example, in our sequence \(a_n = \frac{1 - 2n}{1 + 2n}\), infinity helped us conclude that as \(n\) increases, the influence of constant terms like \(1\) diminishes, leaving the dominant terms to define the behavior, leading to convergence at \(-1\).
Understanding infinity enables us to grasp the eventual outcome of sequences, helping us predict whether they converge to a specific value or not.

Behavior of Functions

Sequences often bear resemblance to functions since sequences map values of \(n\) to real numbers, just like functions do with \(x\). Checking the behavior of sequences is important when analyzing convergence or divergence. This behavior is determined by evaluating the structure of the terms.
When studying functions and sequences:

• Look at the leading terms as \(n\) becomes very large. These dictate how the terms will behave.
• For polynomial expressions, higher degree terms become more influential as \(n\) grows.
• The ratio of coefficients in leading terms often gives the limiting value for rational expressions, as seen in \(a_n = \frac{1-2n}{1+2n}\), where \(\frac{-2}{2}\) simplifies to \(-1\).

By understanding the behavior of functions represented by sequences, one can predict long-term trends and tendencies. Analyzing these behaviors ensures a good grasp of whether the sequence will converge to a limit or diverge.
Recognizing these patterns not only helps in solving sequences but also finds application in calculus and real-world problems where predictions of behavior are essential.