Problem 47 - Thomas Calculus

Question

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=\frac{1}{3}, \quad a_{n+1}=\frac{3 n-1}{2 n+5} a_{n}$$}$$

Step-by-Step Solution

Answer

The series diverges, as the terms do not approach zero.

Step 1: Understand the Problem

We have a series defined by a sequence where the first term is \(a_1 = \frac{1}{3}\) and the recursive formula \(a_{n+1} = \frac{3n - 1}{2n + 5}a_n\). We need to determine whether the sum \(\sum_{n=1}^{\infty} a_n\) converges or diverges.

Step 2: Analyze the Recurrence Relation

The given recurrence relation is \(a_{n+1} = \frac{3n - 1}{2n + 5}a_n\). This tells us how each term in the sequence is derived from the previous one.

Step 3: Investigate Limit of Terms

To investigate the behavior of \(a_n\) as \(n\to\infty\), consider the ratio \(\frac{3n-1}{2n+5}\). As \(n\to\infty\), this ratio tends to \(\frac{3}{2}\), which is greater than 1. This suggests that the terms \(a_n\) do not approach 0.

Step 4: Test for Convergence

For a series \(\sum_{n=1}^{\infty} a_n\) to converge, \(a_n\) must converge to 0. Given that \(\lim_{n \to \infty} a_n eq 0\), the series cannot converge by the divergence test.

Step 5: Conclusion on Series Behavior

Since the terms do not approach zero and instead seem to grow because \(\frac{3}{2} > 1\), the series \(\sum_{n=1}^{\infty} a_n\) is divergent due to the divergence test.

Key Concepts

Recursive Sequences

Recursive sequences are mathematical sequences where each term is defined based on the previous one. In the context of our exercise, the sequence starts with a given initial term \(a_1 = \frac{1}{3}\), and every subsequent term \(a_{n+1}\) is determined by a specific recursive formula: \(a_{n+1} = \frac{3n - 1}{2n + 5}a_n\). Such formulas allow us to precisely define how each term evolves from the preceding one.

Recursive sequences have a wide range of applications and can model numerous real-world processes. Their iterative nature makes them a powerful tool in mathematical analysis and computer science. The main challenge often involves determining the long-term behavior of these sequences as they progress.

• Analyzing the recursive formula helps us determine if the sequence grows, shrinks, or stabilizes.
• Understanding the recursion can provide insights into whether the overall series converges or diverges.

Recognizing recursive relations is crucial because they set the groundwork for further analysis, such as performing a divergence test or finding the limit of sequences.

Divergence Test

The divergence test is a fundamental tool in identifying whether a given infinite series converges or diverges. The test is straightforward: if the limit of the sequence's terms does not approach zero, the series cannot converge. For our sequence \(a_n\), we apply this test to make a conclusion.

In our original exercise, we observe that the ratio \( \frac{3n-1}{2n+5} \) approaches \(\frac{3}{2}\) as \(n\) becomes very large. This significant insight implies that the sequence’s terms do not tend towards zero. Since \(\frac{3}{2} > 1\), terms are actually growing.

• If \ \lim_{n \to \infty} a_n \ eq 0, the series \(\sum_{n=1}^{\infty} a_n\) diverges.
• This conclusion arises because for convergence, the terms must diminish increasingly towards zero.

This test is a critical filter in discerning the nature of infinite series, preventing further analysis when it's clear at the outset that convergence is not possible.

Limit of Sequences

The concept of limits is central to understanding the behavior of sequences as they progress towards infinity. In the context of our exercise, examining the limit informs us about the series' long-term behavior as the number of terms grows.

For the sequence \(a_n\) determined by \(a_{n+1} = \frac{3n - 1}{2n + 5}a_n\), calculating the limit \( \lim_{n \to \infty} \frac{3n-1}{2n+5} \) reveals essential trends. As \(n\) approaches infinity, this limit becomes \(\frac{3}{2}\), which clearly indicates that the terms are not approaching zero. Such a finding is crucial because it directly impacts the convergence of the series.

• Limits provide a mechanism to understand sequences' end behaviors.
• If a sequence does not tend towards zero, the associated series diverge.

Thus, the limit concept assists us in visualizing the sequence's behavior over time, providing invaluable insights into whether the infinite summation of its terms leads to a finite value.