Convergence Analysis Of A Series Defined By A Recurrence Relation

In the fascinating realm of mathematical analysis, the convergence and divergence of infinite series stand as pivotal concepts. Determining whether an infinite series converges to a finite sum or diverges to infinity is a fundamental problem that has captivated mathematicians for centuries. This article delves into the convergence analysis of a specific series defined by a recurrence relation. We are given that the first term of the sequence is ${ a_1 = \frac{1}{3} }$, and the subsequent terms are governed by the recurrence relation ${ a_n = \frac{3n-1}{2n+5} a_{n+1} }$. Our mission is to ascertain whether the series ${ \sum a_n }$ converges or diverges. To embark on this exploration, we will leverage powerful tools from calculus and analysis, including the Ratio Test, to unravel the convergence behavior of this intriguing series.

The heart of our problem lies in the recurrence relation ${ a_n = \frac{3n-1}{2n+5} a_{n+1} }$. This equation elegantly connects each term of the sequence to its successor. To effectively analyze the series ${ \sum a_n }$, we must first gain a deep understanding of how this recurrence relation shapes the sequence ${ \{a_n\} }$. One key step is to rewrite the recurrence relation to express ${ a_{n+1} }$ in terms of ${ a_n }$. This transformation will pave the way for us to apply convergence tests like the Ratio Test. By isolating ${ a_{n+1} }$, we obtain ${ a_{n+1} = \frac{2n+5}{3n-1} a_n }$. This form is particularly insightful because it reveals how each term is scaled relative to its predecessor. The fraction ${ \frac{2n+5}{3n-1} }$ acts as a magnifying or diminishing factor, influencing the growth or decay of the sequence. As ${ n }$ grows large, this fraction approaches ${ \frac{2}{3} }$, suggesting a potential geometric decay behavior. This observation hints that the series might converge, but a rigorous analysis is essential to confirm this intuition.

The Ratio Test is a cornerstone of convergence analysis, providing a powerful criterion for determining the convergence or divergence of infinite series. This test hinges on examining the limit of the ratio of consecutive terms. For a series ${ \sum a_n }$, the Ratio Test considers the limit

${ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| }$

The convergence behavior is then dictated by the value of ${ L }$:

• If ${ L < 1 }$, the series converges absolutely.
• If ${ L > 1 }$, the series diverges.
• If ${ L = 1 }$, the test is inconclusive.

To apply the Ratio Test to our series, we need to compute the ratio ${ \frac{a_{n+1}}{a_n} }$. From our recurrence relation, we know that ${ a_{n+1} = \frac{2n+5}{3n-1} a_n }$. Thus,

${ \frac{a_{n+1}}{a_n} = \frac{2n+5}{3n-1} }$

Now, we take the limit as ${ n }$ approaches infinity:

${ L = \lim_{n \to \infty} \left| \frac{2n+5}{3n-1} \right| }$

To evaluate this limit, we divide both the numerator and the denominator by ${ n }$:

${ L = \lim_{n \to \infty} \left| \frac{2 + \frac{5}{n}}{3 - \frac{1}{n}} \right| }$

As ${ n }$ tends to infinity, the terms ${ \frac{5}{n} }$ and ${ \frac{1}{n} }$ approach zero. Consequently, the limit simplifies to

${ L = \left| \frac{2}{3} \right| = \frac{2}{3} }$

Since ${ L = \frac{2}{3} < 1 }$, the Ratio Test definitively tells us that the series ${ \sum a_n }$ converges absolutely.

Beyond the rigorous application of the Ratio Test, it's insightful to develop an intuitive understanding of why this series converges. The key lies in the behavior of the ratio ${ \frac{2n+5}{3n-1} }$ as ${ n }$ grows large. As we observed earlier, this ratio approaches ${ \frac{2}{3} }$ for large ${ n }$. This implies that, for sufficiently large ${ n }$, each term ${ a_{n+1} }$ is approximately ${ \frac{2}{3} }$ times its predecessor ${ a_n }$. This geometric decay is a hallmark of convergent series. In essence, the terms of the sequence are shrinking rapidly enough to ensure that their sum remains finite. We can compare this behavior to a geometric series with a common ratio of ${ \frac{2}{3} }$, which is known to converge. While our series is not strictly geometric, its asymptotic behavior mirrors that of a convergent geometric series, providing an intuitive justification for its convergence.

In this article, we have embarked on a comprehensive analysis of the series ${ \sum a_n }$, where the sequence ${ \{a_n\} }$ is defined by the recurrence relation ${ a_1 = \frac{1}{3} }$ and ${ a_n = \frac{3n-1}{2n+5} a_{n+1} }$. By rewriting the recurrence relation, we were able to express ${ a_{n+1} }$ in terms of ${ a_n }$, paving the way for the application of the Ratio Test. The Ratio Test, a cornerstone of convergence analysis, decisively revealed that the series converges absolutely. Furthermore, we delved into an intuitive understanding of the convergence, emphasizing the geometric decay behavior of the terms. The ratio of consecutive terms approaches ${ \frac{2}{3} }$ as ${ n }$ grows large, mirroring the behavior of a convergent geometric series. This exploration underscores the power of analytical tools in unraveling the convergence behavior of infinite series, providing a glimpse into the rich tapestry of mathematical analysis. The convergence of this series serves as a testament to the intricate interplay between recurrence relations and the convergence of infinite sums.

The series ${ \sum a_n }$ converges.