Convergence Test For The Series ∑ (n^(3/2)) Divided By (4n^3 - 3n^2)

Understanding Series Convergence

In the realm of mathematical analysis, determining the convergence or divergence of an infinite series is a fundamental task. A series, represented as the sum of an infinite sequence of terms, converges if the sequence of its partial sums approaches a finite limit. Conversely, if the partial sums do not approach a finite limit, the series diverges. Understanding the behavior of infinite series is crucial in various fields, including calculus, differential equations, and mathematical physics. Several tests are available to ascertain the convergence or divergence of a given series, each with its own set of applicability and limitations. These tests provide the tools necessary to analyze the long-term behavior of infinite sums and their relevance in mathematical models and applications. The significance of convergence tests lies in their ability to provide rigorous methods for analyzing the behavior of infinite sums, ensuring that mathematical models and computations are well-defined and reliable. By applying these tests, mathematicians and scientists can confidently work with infinite series, knowing whether the sum will approach a finite value or grow without bound. This understanding is essential for developing accurate models and making sound predictions in various fields. In particular, determining whether a series converges is essential in applications involving approximations, such as numerical analysis and the development of algorithms. If a series diverges, using it for approximations can lead to incorrect or nonsensical results, highlighting the importance of convergence testing.

The Series in Question: ∑ (n^(3/2)) / (4n^3 - 3n^2)

Our focus is on the series defined by the summation ∑ (n^(3/2)) / (4n^3 - 3n^2), where the summation is taken over positive integers n. This series presents a rational function of n within the summand, making it a suitable candidate for analysis using various convergence tests. The numerator of the summand is n^(3/2), a power function that grows polynomially with n. The denominator, 4n^3 - 3n^2, is a cubic polynomial in n. As n becomes very large, the term 4n^3 dominates the denominator's behavior. To rigorously test this series for convergence, we must employ appropriate techniques from the arsenal of convergence tests. These tests offer different approaches to examine the behavior of the series as n approaches infinity. The choice of test often depends on the specific structure of the series, and some tests may be more effective than others in certain cases. In our analysis, we'll explore the application of the limit comparison test, which is particularly well-suited for series where the terms can be compared to a simpler, known series. This approach allows us to determine whether the given series converges or diverges by relating its behavior to that of a series whose convergence properties are already established. By carefully applying this test, we can gain a clear understanding of the convergence behavior of the series in question and its implications in mathematical contexts.

Applying the Limit Comparison Test

The limit comparison test is a powerful tool for determining the convergence or divergence of a series by comparing it to another series with known behavior. This test is particularly useful when dealing with series whose terms can be related to simpler terms as n approaches infinity. To apply the limit comparison test to our series, ∑ (n^(3/2)) / (4n^3 - 3n^2), we need to select a suitable comparison series. As n becomes large, the term 3n^2 in the denominator becomes insignificant compared to 4n^3. Therefore, the summand behaves approximately like n^(3/2) / (4n^3), which simplifies to 1 / (4n^(3/2)). This suggests that we can compare our series to the simpler series ∑ 1 / n^(3/2). This series is a p-series with p = 3/2. A p-series is a series of the form ∑ 1 / n^p, and it is known to converge if p > 1 and diverge if p ≤ 1. Since 3/2 > 1, the series ∑ 1 / n^(3/2) converges. Now, we need to formally apply the limit comparison test. We consider the limit as n approaches infinity of the ratio of the terms of our series to the terms of the comparison series:

lim (n→∞) [(n^(3/2)) / (4n^3 - 3n^2)] / [1 / n^(3/2)]

This simplifies to:

lim (n→∞) [n^3 / (4n^3 - 3n^2)]

Dividing both the numerator and the denominator by n^3, we get:

lim (n→∞) [1 / (4 - 3/n)]

As n approaches infinity, 3/n approaches 0, so the limit becomes:

1 / (4 - 0) = 1/4

Since the limit is a finite positive number (1/4), the limit comparison test tells us that our original series, ∑ (n^(3/2)) / (4n^3 - 3n^2), behaves in the same way as the comparison series, ∑ 1 / n^(3/2). Because the comparison series converges (as it is a p-series with p = 3/2 > 1), we conclude that our series also converges. This result provides a rigorous confirmation of the convergence behavior of the series, which is crucial in mathematical analysis and applications.

Conclusion: Convergence of the Series

Based on the application of the limit comparison test, we have conclusively demonstrated that the series ∑ (n^(3/2)) / (4n^3 - 3n^2) converges. The key to this determination was the comparison with the p-series ∑ 1 / n^(3/2), which is known to converge because its exponent p = 3/2 is greater than 1. The limit comparison test allowed us to rigorously relate the behavior of our series to that of the p-series, establishing a clear connection between their convergence properties. This result is significant because it provides a solid foundation for further analysis or application of this series in mathematical models or computations. Convergence ensures that the sum of the infinite series approaches a finite value, making it a meaningful quantity to work with. If the series had diverged, the sum would grow without bound, potentially leading to inconsistencies or inaccuracies in calculations. The convergence of this particular series highlights the importance of carefully selecting and applying appropriate convergence tests. Different tests may be more suitable for different types of series, and the limit comparison test proved to be a powerful tool in this case due to the rational function structure of the summand. The ability to accurately determine the convergence of series is a cornerstone of mathematical analysis, enabling mathematicians and scientists to work confidently with infinite sums and their applications in various fields.

In summary, the series ∑ (n^(3/2)) / (4n^3 - 3n^2) converges, as demonstrated by the limit comparison test with the convergent p-series ∑ 1 / n^(3/2). This result underscores the importance of convergence testing in mathematical analysis and its applications.