Problem 26
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
Verified Answer
The sequence is convergent, and its limit is 0.
1Step 1: Identify the Sequence
The given sequence is \(a_n = \frac{n-1}{n^3+1}\). We need to determine whether this sequence converges, and if so, find its limit.
2Step 2: Examine Behavior as n Approaches Infinity
To analyze the limit as \(n\) approaches infinity, consider the dominant terms in both the numerator and the denominator. For large \(n\), the dominant term in the numerator (\(n-1\)) is \(n\) and in the denominator (\(n^3+1\)) is \(n^3\).
3Step 3: Simplified Fraction for Large n
For large \(n\), approximate \(a_n\) by simplifying the dominant terms: \(a_n \approx \frac{n}{n^3} = \frac{1}{n^2}\).
4Step 4: Determine the Limit
Since \(\frac{1}{n^2}\) approaches 0 as \(n\) tends to infinity, the limit of the sequence \(a_n\) is 0.
5Step 5: Conclusion of Convergence
Since the limit of the sequence \(a_n = \frac{n-1}{n^3+1}\) is 0, the sequence is convergent.
Key Concepts
Limit of a SequenceDominant TermsInfinite Sequences
Limit of a Sequence
The limit of a sequence is a foundational concept in calculus and real analysis. When we talk about the limit of a sequence, we refer to the value that the elements of a sequence approach as the index (often denoted as \( n \)) goes to infinity. Think of it like asking what number the sequence "settles down to" as you keep moving along—the long-term behavior of the sequence.
To find the limit, one might inspect the formula of the sequence, which often involves examining the behavior as \( n \) becomes very large. Just like in our exercise, we deduced that the given sequence \( a_n = \frac{n-1}{n^3+1} \) converges to the limit of 0. This is because the terms simplify to \( \frac{1}{n^2} \) which clearly approaches 0 as \( n \to \infty \).
The notion of a limit helps in determining whether a sequence is convergent or divergent. If a sequence has a finite limit, it is considered convergent. Otherwise, it is divergent.
To find the limit, one might inspect the formula of the sequence, which often involves examining the behavior as \( n \) becomes very large. Just like in our exercise, we deduced that the given sequence \( a_n = \frac{n-1}{n^3+1} \) converges to the limit of 0. This is because the terms simplify to \( \frac{1}{n^2} \) which clearly approaches 0 as \( n \to \infty \).
The notion of a limit helps in determining whether a sequence is convergent or divergent. If a sequence has a finite limit, it is considered convergent. Otherwise, it is divergent.
Dominant Terms
Dominant terms play a critical role in analyzing the behavior of sequences, especially when we seek to determine convergence or divergence. When we refer to dominant terms, we are talking about the parts of an expression that have the greatest impact as \( n \) becomes very large or very small.
In our exercise, for \( a_n = \frac{n-1}{n^3+1} \), as \( n \) grows, \( n \) becomes dominant in the numerator, and \( n^3 \) dominates in the denominator. This means that these terms will majorly determine the value of \( a_n \) for large \( n \).
By simplifying these dominant terms, we can often see patterns or simpler expressions that lead us directly to the sequence's behavior at infinity. This concept is not only useful for finding the limit of a sequence but also for comparing different sequences to see which one grows faster or slower.
In our exercise, for \( a_n = \frac{n-1}{n^3+1} \), as \( n \) grows, \( n \) becomes dominant in the numerator, and \( n^3 \) dominates in the denominator. This means that these terms will majorly determine the value of \( a_n \) for large \( n \).
By simplifying these dominant terms, we can often see patterns or simpler expressions that lead us directly to the sequence's behavior at infinity. This concept is not only useful for finding the limit of a sequence but also for comparing different sequences to see which one grows faster or slower.
Infinite Sequences
Infinite sequences are sequences that continue indefinitely, without end. Each term in a sequence is indexed by an integer, and it's important to consider what happens to the terms as the index approaches infinity.
In mathematical terms, an infinite sequence \( \{a_n\} \) is a function whose domain is the set of natural numbers, and mapping these numbers to a set of terms. Often what interests us the most is whether the sequence converges to a particular limit or behaves chaotically without settling down to any value.
In our example, the sequence \( a_n = \frac{n-1}{n^3+1} \) is infinite because it carries on as long as there are integers, \( n \).
Infinite sequences can model an array of phenomena in the real world, ranging from simple arithmetic progressions to more complex scenarios like adjusting for random variance in data. Understanding how these sequences behave is crucial in fields like calculus, algorithms, and physics.
In mathematical terms, an infinite sequence \( \{a_n\} \) is a function whose domain is the set of natural numbers, and mapping these numbers to a set of terms. Often what interests us the most is whether the sequence converges to a particular limit or behaves chaotically without settling down to any value.
In our example, the sequence \( a_n = \frac{n-1}{n^3+1} \) is infinite because it carries on as long as there are integers, \( n \).
Infinite sequences can model an array of phenomena in the real world, ranging from simple arithmetic progressions to more complex scenarios like adjusting for random variance in data. Understanding how these sequences behave is crucial in fields like calculus, algorithms, and physics.
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