Problem 34 - 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. $$ a_{n}=\frac{1-n^{3}}{70-4 n^{2}} $$

Step-by-Step Solution

Answer

The sequence diverges because it approaches infinity as \( n \to \infty \).

Step 1: Check the form of the sequence

First, analyze the given sequence: \( a_n = \frac{1-n^3}{70-4n^2} \). Both the numerator and the denominator are polynomials, with the highest power being in the numerator: \( n^3 \) versus \( n^2 \) in the denominator.

Step 2: Determine the dominant term

For large \( n \), the dominant term in the numerator is \( -n^3 \), and the dominant term in the denominator is \( -4n^2 \). This simplifies the sequence to \( a_n \approx \frac{-n^3}{-4n^2} = \frac{n}{4} \).

Step 3: Analyze the behavior as n approaches infinity

As \( n \to \infty \), the reduced sequence \( \frac{n}{4} \to \infty \). Therefore, the sequence \( a_n \) diverges because it approaches infinity, not a finite limit.

Key Concepts

Convergence and Divergence

Understanding convergence and divergence is essential when analyzing sequences. A sequence converges if it approaches a specific, finite number as the terms continue indefinitely. Conversely, a sequence diverges if it does not settle at any particular value, possibly growing towards infinity or oscillating without end.

Consider the sequence provided: \( a_n = \frac{1-n^3}{70-4n^2} \). At first glance, to determine whether it converges or diverges, we compare the behaviors of both the numerator and denominator for large values of \( n \). This form, common in fractions of polynomials, requires careful analysis of the dominant terms.

If the numerator's power is greater than the denominator's, as it is here (\( n^3 \) versus \( n^2 \)), the sequence tends to diverge. Thus, when the significant term (here \( n^3 \)) overpowers the rest, the sequence will not reach a finite limit, leading to divergence. The convergence analysis shows that as \( n \to \infty \), the sequence \( a_n \) does not converge to a specific value but instead diverges because it approaches infinity.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence get closer to, as the term number increases to infinity. If a sequence converges, it has a limit; if it diverges, it does not settle on any one number.

For the sequence \( a_n = \frac{1-n^3}{70-4n^2} \), we examine the behavior of the sequence for large values of \( n \). Observing the polynomial powers, the highest power term \( n^3 \) in the numerator affects the sequence's growth rate. To find the limit, it's practical to simplify using the dominant term analysis, resulting in \( \frac{n^3}{4n^2} = \frac{n}{4} \).

As \( n \) approaches infinity, instead of reaching a fixed number, \( \frac{n}{4} \) itself approaches infinity, indicating there is no finite limit. Consequently, this sequence lacks a limit because the terms grow indefinitely rather than approaching a specific value.

Dominant Term Analysis

Dominant term analysis is a technique used to simplify the understanding of sequences, especially when they involve polynomials. It involves identifying the term in both the numerator and denominator with the highest power of \( n \), as this term dictates the sequence's behavior.

In the original sequence \( a_n = \frac{1-n^3}{70-4n^2} \), the dominant terms are \(-n^3\) in the numerator and \(-4n^2\) in the denominator. By concentrating on these terms, we approximate the sequence's behavior for large \( n \) as \( \frac{n}{4} \). This approximation simplifies the analysis, allowing us to predict whether the sequence diverges or converges.

This method is invaluable for polynomial sequences where other lower power terms become negligible as \( n \) grows. By focusing on the key terms, we accurately determine the sequence's long-term tendencies and find that the approximation suggests a divergence, confirming that the sequence \( a_n \) grows without bound.