Problem 31 - 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. $$ a_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{3}} $$

Step-by-Step Solution

Answer

The sequence converges to -5.

Step 1: Recognize the Dominant Terms

In the expression \(a_n = \frac{1-5n^4}{n^4+8n^3}\), the highest power of \(n\) in both the numerator and the denominator is \(n^4\). Analyzing this will help us understand the behavior of the sequence as \(n\) approaches infinity.

Step 2: Simplify the Fraction by Dominant Terms

Since \(n^4\) is the highest power, divide both the numerator and the denominator by \(n^4\) to simplify: \[a_n = \frac{\frac{1}{n^4} - 5}{1 + \frac{8}{n}}\].

Step 3: Evaluate the Limits of Each Term

As \(n\) approaches infinity, \(\frac{1}{n^4} \rightarrow 0\) and \(\frac{8}{n} \rightarrow 0\). Substitute these limits into the simplified fraction: \[a_n = \frac{0 - 5}{1 + 0} = -5\].

Step 4: Conclusion on Convergence and Limit

The sequence \(a_n = \frac{1-5n^4}{n^4+8n^3}\) converges to \(-5\) as \(n\) approaches infinity.

Key Concepts

Limits

In the context of sequences, a limit refers to the value that a sequence approaches as the index becomes indefinitely large. Understanding limits is crucial for analyzing if sequences converge or diverge. For any sequence \(a_n\), you want to determine \(L\) such that, for every positive number \(\epsilon\), there exists a positive integer \(N\) where if \(n > N\), then \(|a_n - L| < \epsilon\). This formal definition shows how close the terms of the sequence get to \(L\) as \(n\) increases.

When handling limits in rational expressions like \(\frac{1-5n^4}{n^4+8n^3}\), scrutinize the behavior of each component as \(n\) approaches infinity. Often, simplification via dominant terms assists in recognizing the limit, making it simpler to see if a sequence converges. In our example, the limit \(-5\) is discerned by realizing that non-dominant parts become negligible.

Dominant Terms

Dominant terms in a sequence or expression are those which significantly affect the behavior, especially when variables are large. In the expression \(a_n = \frac{1 - 5n^4}{n^4 + 8n^3}\), the dominant term in the numerator is \(-5n^4\) and in the denominator is \(n^4\). These terms "dominate" because they will have the most considerable impact on the expression's value as n becomes very large.

To analyze such sequences, factor out the dominant terms. Here, you divide both the numerator and the denominator by \(n^4\), yielding \(a_n = \frac{\frac{1}{n^4} - 5}{1 + \frac{8}{n}}\). As \(n\) increases, terms like \(\frac{1}{n^4}\) and \(\frac{8}{n}\) tend to zero, highlighting the dominance of what remains. This process simplifies the evaluation of limits and clarifies the sequence's behavior without unnecessary complexity.

Infinity Behavior

Analyzing the infinity behavior of a sequence involves understanding how the terms behave as \(n\) becomes extremely large or tends toward infinity. Knowing this helps distinguish between convergent and divergent sequences. For the sequence \(a_n = \frac{1-5n^4}{n^4+8n^3}\), by observing infinity behavior, you simplify and conclude that most terms vanish against the dominant ones as \(n\) grows.

In the case of \(\frac{1-5n^4}{n^4+8n^3}\), the dominant terms are \(n^4\) for both the numerator and the denominator. As n approaches infinity:

• \(\frac{1}{n^4}\) approaches 0
• \(\frac{8}{n}\) approaches 0

This allows the sequence to be approximated as \(a_n = \frac{-5}{1} = -5\), meaning it converges to \(-5\). Understanding infinity behavior thus helps in determining the fate of sequences.