Problem 26 - Precalculus: Mathematics for Calculus

Question

Limits of Sequences If the sequence with the given \(n\) th term 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: Understanding the Sequence Limit

We need to find the limit of the sequence given by \( a_n = \frac{n-1}{n^3+1} \) as \( n \to \infty \). This involves analyzing the behavior of the sequence term \( a_n \) as \( n \) becomes very large.

Step 2: Simplifying the Sequence

The dominant term in the denominator is \( n^3 \) as \( n \to \infty \), so we can approximate the denominator as \( n^3 \). Thus, \( a_n \approx \frac{n-1}{n^3} = \frac{n}{n^3} - \frac{1}{n^3} = \frac{1}{n^2} - \frac{1}{n^3} \).

Step 3: Analyzing Simplified Expression

As \( n \to \infty \), both \( \frac{1}{n^2} \) and \( \frac{1}{n^3} \) tend to 0 because any positive power of \( n \) in the denominator will tend towards zero. This indicates that \( a_n \to 0 \).

Step 4: Concluding the Sequence Behavior

Since both parts of the sequence expression tend to zero as \( n \to \infty \), the sequence \( a_n = \frac{n-1}{n^3+1} \) is convergent and the limit of the sequence is \( 0 \).

Key Concepts

Convergence

Convergence in sequences refers to the behavior of a sequence as its terms get closer to a specific value as the index approaches infinity. When we say a sequence converges, it means that as you take more and more terms of the sequence, the terms get closer and closer to a particular number. In the given sequence, \( a_n = \frac{n-1}{n^3+1} \), convergence is determined by evaluating the expression as \( n \) becomes very large. As explained in the solution, this sequence simplifies to an expression where both components approach zero, leading to a limit of 0. This confirms that the sequence is convergent. Convergence is a crucial concept because it helps us understand the long-term behavior of sequences, informing whether a consistent pattern or fixed value emerges.

Sequence Analysis

Analyzing sequences involves understanding their patterns and behavior. To analyze the given sequence \( a_n = \frac{n-1}{n^3+1} \), we look for the dominant terms. The trick lies in simplifying the sequence to make it easier to evaluate as \( n \) grows infinitely large. Here, the *numerator* is \( n-1 \) and the *denominator* is \( n^3 + 1 \).

• The term \( n^3 \) in the denominator becomes the dominant term as \( n \) increases since cubic terms grow faster than linear terms like \( n \) or constants like 1.
• By approximating the denominator with \( n^3 \), the sequence simplifies to \( \frac{n}{n^3} - \frac{1}{n^3} = \frac{1}{n^2} - \frac{1}{n^3} \).

This kind of breakdown allows you to compute the behavior of the sequence efficiently, revealing that both terms approach zero individually, thus making sequence analysis crucial for determining convergence.

Infinite Behavior

The infinite behavior of a sequence describes what happens to the terms of a sequence as we continue to larger and larger values of \( n \). When we consider the sequence \( a_n = \frac{n-1}{n^3+1} \), analyzing its infinite behavior helps us understand how the sequence progresses as \( n \to \infty \).
In this example, simplifying the sequence shows that as \( n \) becomes very large:

• Both \( \frac{1}{n^2} \) and \( \frac{1}{n^3} \) decrease towards zero because of the increasing power in the denominator.
• This diminishing tendency means that each term in the sequence gets closer to zero, indicating a behavior of convergence to 0 as \( n \to \infty \).

The discussion of infinite behavior is essential because it addresses how sequences behave in the limit. Such insight is valuable for predicting long-term outcomes and ensuring precise mathematical understanding of sequences.