Problem 38 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-n^{3}}{70-4 n^{2}} $$

Step-by-Step Solution

Answer

The sequence diverges; it does not converge to a finite limit.

Step 1: Analyze the Sequence Behavior

To determine whether the sequence \(a_n = \frac{1-n^3}{70-4n^2}\) converges or diverges, start by analyzing the expression's behavior as \(n \to \infty\). Observe that in the numerator, the term \(-n^3\) will dominate, and in the denominator, \(-4n^2\) will dominate for large \(n\).

Step 2: Simplify the Sequence

Simplify the sequence by dividing both the numerator and denominator by \(n^3\), the highest power of \(n\) in the function: \[a_n = \frac{\frac{1}{n^3} - 1}{\frac{70}{n^3} - \frac{4}{n}}\].

Step 3: Evaluate the Limit as n Approaches Infinity

Evaluate the limit \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{\frac{1}{n^3} - 1}{\frac{70}{n^3} - \frac{4}{n}}\). As \(n\) approaches infinity, \(\frac{1}{n^3} \to 0\) and \(\frac{70}{n^3} \to 0\), leaving \(\lim_{n \to \infty} a_n = \frac{0 - 1}{0 - 0} = -\infty\).

Step 4: Conclude on Convergence and Limit

Since the limit of \(a_n\) as \(n\) approaches infinity does not exist as a finite number, the sequence diverges.

Key Concepts

Sequence Convergence

Understanding whether a sequence converges is essential in mathematics as it tells us whether the sequence approaches a specific value as it progresses. For a sequence to converge, its terms must get closer and closer to a particular point as the number of terms increases. This "target" point or value is known as the limit.
In the context of our exercise with the sequence \(a_n = \frac{1-n^3}{70-4n^2}\), we start by examining the behavior of the sequence's terms as \(n\) becomes very large. Convergence implies that as \(n\) tends to infinity, the sequence heads towards one specific value. Therefore, our task was to determine if \(a_n\) settles ton a fluid endpoint or continues indefinitely apart. In this case, after analysis, we concluded that the sequence diverges, meaning it does not stabilize towards a specific finite limit.

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the sequence progresses. When studying limits, we want to determine what number, if any, the terms of the sequence increasingly become arbitrarily close to.
In our exercise involving \(a_n = \frac{1-n^3}{70-4n^2}\), finding the limit means plugging extremely large values of \(n\) into our formula to see what value \(a_n\) approaches. We determine the limit analytically by simplifying the sequence and looking at the dominant terms that affect behavior at infinity.

• For the given sequence, the limit was calculated by dividing the numerator and denominator by \(n^3\), the highest power.
• Consequently, as \(n\) tends to infinity, it became evident that both \(\frac{1}{n^3}\) and \(\frac{70}{n^3}\) move towards 0, leaving us with \(\frac{0-1}{0-0} = -\infty\).

This reveals the divergence of the sequence since a finite limit does not exist.

Asymptotic Behavior

Asymptotic behavior refers to the tendency of a sequence or function as its variable - in this case, \(n\) - becomes very large. Here, it shows how the sequence behaves at the "boundary" at infinity.
When assessing \(a_n = \frac{1-n^3}{70-4n^2}\), we analyzed the terms within the expression to understand which would influence its behavior most prominently as \(n\) grows extensively. We identified that \(-n^3\) and \(-4n^2\) are the dominant terms in the numerator and denominator respectively.

• The term \(-n^3\) affects the direction and rate of overall sequence growth or decline.
• Meanwhile, \(-4n^2\) dictates the decline in the denominator.

Identifying which terms dominate as \(n\) becomes large is crucial in predicting asymptotic behavior. In our example, these led to an ever-increasing negative fraction, indicating divergence towards \(-\infty\).

Dominant Term Analysis

Dominant term analysis is a mathematical technique used to understand the behavior of sequences or functions for large values of variables by focusing on the largest power terms because they significantly influence the form of the sequence.
When applying this technique to the sequence \(a_n = \frac{1-n^3}{70-4n^2}\), we targeted the highest power of \(n\) in both the numerator and the denominator:

• In the numerator, \(-n^3\) was the dominant term.
• In the denominator, \(-4n^2\) was key.

By dividing all terms by the highest power of \(n\) from the numerator, we were able to simplify the analysis of the sequence’s behavior as \(n\) approaches infinity. It clarified that residual terms become negligible, and thus these dominant components dictate overall growth, aiding in determining the sequence's limit or divergence. This ensures precision when predicting the behavior of sequences for large \(n\).