Problem 22 - Precalculus: Mathematical for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n-1}{n^{3}+1}$$

Step-by-Step Solution

Answer

The sequence is convergent with a limit of 0.

Step 1: Identify the Sequence

The sequence given is \( a_n = \frac{n-1}{n^3+1} \). We need to determine if this sequence is convergent and, if so, find its limit.

Step 2: Analyze the Behavior as \( n \to \infty \)

As \( n \) approaches infinity, the highest degree term in the numerator is \( n \), and in the denominator, it is \( n^3 \). Therefore, the behavior is dominated by these terms.

Step 3: Simplify the Terms

Divide both the numerator and the denominator by \( n^3 \), the highest power of \( n \) in the denominator: \[ a_n = \frac{\frac{n}{n^3} - \frac{1}{n^3}}{\frac{n^3}{n^3} + \frac{1}{n^3}} = \frac{\frac{1}{n^2} - \frac{1}{n^3}}{1 + \frac{1}{n^3}}. \]

Step 4: Evaluate the Limit

As \( n \to \infty \), \( \frac{1}{n^2} \to 0 \), \( \frac{1}{n^3} \to 0 \), and the sequence becomes: \[ a_n \approx \frac{0 - 0}{1 + 0} = 0. \] Thus, the limit of the sequence \( a_n \) is 0.

Key Concepts

Sequence Limit

The sequence limit is the value a sequence approaches as the number of terms goes to infinity. When we talk about whether a sequence is convergent, we're essentially asking if it reaches a fixed limit. For the given sequence \(a_n = \frac{n-1}{n^3+1} \), we'd like to calculate what the sequence does as \(n\) gets larger and larger. If it approaches a specific number, then it converges, and that number is the sequence's limit.
To determine the limit, we focus on dividing both the numerator and denominator by the highest degree term in the denominator, which simplifies the expression. This will help us see clearly what number, if any, the sequence approaches. If a fraction's terms approach zero, then the sequence converges to the calculated fraction, which is often zero in cases of dominance by the denominator as observed here.

Infinity Behavior

Infinity behavior describes what happens to the sequence as the terms grow infinitely large—meaning as \(n\) approaches infinity. As we consider \(n\) getting larger, it’s key to identify which terms dictate the behavior. In \(a_n = \frac{n-1}{n^3+1}\), the numerator’s behavior, dominated by \(n\), increases linearly, while the denominator’s behavior depends primarily on \(n^3\), increasing much more rapidly.
So, as \(n\) increases, the term \(n^3\) becomes significantly larger than the linear \(n\) in the numerator. Thus, the entire fraction tends towards zero because the denominator grows faster than the numerator. This analysis helps us to conclude that the sequence converges towards its limit, due to the overpowering growth of the highest degree term in the denominator.

Highest Degree Term

The highest degree term is crucial in determining how each part of a fraction behaves as \(n\) becomes very large. In calculus and sequences analysis, this term mainly reveals which part of the fraction will dominate as \(n\) increases to infinity. For the provided sequence, \(a_n = \frac{n-1}{n^3+1}\), we check both the numerator and the denominator.

• Numerator's highest degree term: \(n\)
• Denominator's highest degree term: \(n^3\)

To simplify, both are divided by \(n^3\), the highest power in the denominator. This leads us to focus on the term \(\frac{1}{n^2}\) in the numerator for large \(n\), which tends to zero as \(n\) goes to infinity. This analysis significantly aids in understanding why the sequence converges to zero, as the influence of the smaller powers vanishes in comparison.