Question

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\)

Short answer

The sequence \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\) converges and its limit is \(\frac{3}{2}\).

Step-by-step solution

Understand the problem

The task is to find the limit of the sequence defined by \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\) as \(n\) approaches infinity. If the limit exists and is a real number, the sequence converges to that number. If the limit is not a real number or does not exist, the sequence diverges.

Find the limit

The limit of a sequence as \(n\) approaches infinity is found by substituting \(n\) with infinity in the expression for the sequence. However, as the \(n^{th}\) term here is a rational function, it's not enough to just substitute \(n\) with infinity. We instead look at the highest powers in the numerator and the denominator. As \(n \to \infty\), the lower order terms go to 0 and the limit of the sequence becomes the coefficient of the highest power terms divided by each other, which here is \(\frac{3}{2}\).

Determine if the sequence converges

As the limit is a finite real number, \(\frac{3}{2}\), the sequence \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\) converges to this limit.

Key concepts

Limits of a Sequence

In the study of mathematics, particularly calculus, understanding the limit of a sequence is a fundamental concept. This involves determining what value a sequence approaches as the term number becomes very large, often referred to as approaching infinity. For the given sequence \(a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}\), we are interested in what happens as \(n\) grows without bounds.\
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To calculate this, we generally look for patterns or use rules for finding limits. One such rule is focusing on the highest power of \(n\) in both the numerator and denominator. In our case, the term \(3n^2\) in the numerator and the term \(2n^2\) in the denominator dominate as \(n\) becomes large because all other terms become less significant. Therefore, the limit of the sequence as \(n\) approaches infinity is simply the ratio of the leading coefficients, which is \(\frac{3}{2}\). The presence of a finite limit suggests that the sequence converges to that specific value, indicating stability in its long-term behavior.\
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In more complex sequences, finding limits can involve additional techniques such as l'Hôpital's rule, recursive formulas, or squeeze theorem. However, in this case, recognizing the dominance of the highest power terms simplified the process considerably.

Convergence and Divergence

When exploring sequences, the terms convergence and divergence are essential indicators of a sequence's behavior. A sequence is said to converge if, as \(n\) increases, the values of the sequence approach a specific number, called the limit. This limit can be visualized as a horizontal line that the terms of the sequence get closer to on a graph as \(n\) goes towards infinity.\
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In contrast, divergence occurs when the sequence doesn't settle towards any particular value. Divergent sequences might oscillate between values, grow without bound, or decrease without approaching a specific value. Determining whether a sequence converges or diverges helps mathematicians and scientists predict the behavior of sequences that model real-world phenomena. In our example, since the limit \(a_n\) as \(n\) goes to infinity is \(\frac{3}{2}\), it indicates that the sequence converges to \(\frac{3}{2}\). This result is a significant finding and can imply stability in the modeled process assuming the sequence represents some real-world variable over time.

Rational Functions

A rational function is a ratio of two polynomials. In our exercise, \(a_n=\frac{3 n^{2}-n+4}{2 n^{2}+1}\) is a rational function where the polynomials in the numerator and the denominator are both of degree 2. Rational functions can be simple or complex, and their behavior can be studied using limits.\
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One way to analyze their long-term behavior, as \(n\) becomes very large or very small, is to consider the end-behaviors of the function. This involves evaluating the highest power terms in both the numerator and the denominator, as we did in the given exercise. In some cases, particularly when the degrees of the numerator and denominator are different, the rational function may approach zero or infinity, or even become undefined. \
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However, when the degrees are the same, as in our example, the long-term behavior is the quotient of the leading coefficients, since the relative size of the other terms diminishes as \(n\) increases. Understanding the behavior of rational functions is key to fields such as engineering, physics, and economics, where modeling dynamic systems is crucial. The rational function in our exercise models a sequence that, after analysis, we found to converge, offering insight into the system's future behavior based on its mathematical properties.