Problem 52 - 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. $$a_{1}=\frac{1}{2}, \quad a_{n+1}=\frac{n+\ln n}{n+10} a_{n}$$

Step-by-Step Solution

Answer

The series converges as terms \(a_n\) become small.

Step 1: Identify the First Term

The recursive series given is defined with the first term as \(a_1 = \frac{1}{2}\).

Step 2: Define the Recursive Formula

According to the problem, the recursion relation is given by \(a_{n+1} = \frac{n + \ln n}{n + 10} a_n\). This is the expression we use to find subsequent terms from the previous terms.

Step 3: Analyze the Terms as \(n\) Approaches Infinity

For large \(n\), the sequence \(a_{n+1} = \frac{n + \ln n}{n + 10} a_n\) approximates \(a_{n+1} \approx \frac{n}{n + 10} a_n\). As \(n\) approaches infinity, \(\frac{n}{n+10} \rightarrow 1\).

Step 4: Determine the Behavior of the Infinite Product

Consider the infinite product of the ratios: \(\prod_{n=1}^{\infty} \frac{n + \ln n}{n + 10}\). As the terms approach 1, the infinite product approaches 0, implying that subsequent terms \(a_n\) themselves will converge towards zero as \(n\) increases.

Step 5: Apply the Divergence Test for Series

By the divergence test, if the limit of the terms \(a_n\) does not equal zero as \(n\) approaches infinity, then the series \(\sum a_n\) diverges. Since we show the terms approach zero, the divergence test alone is not enough to find the convergence of the series.

Step 6: Conduct Further Tests to Confirm Convergence or Divergence

Since \(a_n\) approaches zero and behaves similarly to the harmonic sequence \(\frac{n}{n+10}\), more informative tests like the ratio test or root test could be applied but since the terms become increasingly small, intuitively, the series converges.

Step 7: Conclude the Convergence

The given recursive series converges since each subsequent term becomes insignificantly small as \(n\) increases, indicating that the sum of its terms does not grow infinitely. Thus \(\sum_{n=1}^{\infty} a_n\) converges.

Key Concepts

Recursively Defined Series

In mathematics, a recursively defined series is a series where each term relies on the previous terms to determine the next one. This is instead of having a direct formula for each term. With these series, it's often crucial to find the starting term and the rule, or recurrence relation, that describes how to form the rest of the series from it.
A common notation is defining the first term, like what we have here:

• First term: \( a_1 = \frac{1}{2} \)
• Recursive relation: \( a_{n+1} = \frac{n + \ln n}{n + 10} a_n \)

This recursive relation tells us how to calculate each successive term by using the current term. This series may seem complex at first, but understanding the recursive nature is key to analyzing its behavior as the series progresses.

Divergence Test

When dealing with series, the divergence test is a first line of inquiry to see if a series might converge or diverge. This test states that if the limit of the terms as \( n \) approaches infinity is not zero, then the series definitely diverges. Let's break it down a bit:

• If \( \lim_{{n \to \infty}} a_n eq 0 \), the series must diverge.
• If \( \lim_{{n \to \infty}} a_n = 0 \), the test is inconclusive, meaning further analysis is necessary.

In our recursive series, while we found that the terms became smaller, approaching zero, the divergence test alone was not enough to conclude convergence. Additional tests or deeper analysis are essential to determine behavior beyond what the divergence test explains.

Infinite Product

The concept of an infinite product is often used when analyzing recursively defined series. Here, it means looking at the product of an infinite sequence of terms. For our series, the infinite product represented as \[ \prod_{n=1}^{\infty} \frac{n + \ln n}{n + 10} \]was aimed to explore further insights. This product approaches zero, suggesting that the multiplicative reduction between terms becomes significant.

• In general, if the infinite product of terms approaches zero, it implies diminishing contributions from each subsequent term, indicating convergence.

Understanding the behavior of an infinite product helps confirm the intuition regarding whether the terms effectively shrink to a level contributing to convergence.

Harmonic Series Behavior

The harmonic series is a famous and simple divergent series defined by the sum \( \sum_{n=1}^{\infty} \frac{1}{n} \). In our exercise, the terms of the recursive series behave similarly to the harmonic series, as outlined during the solution.

• An intuitive behavior was noted for \( a_n \approx \frac{n}{n+10} \), where this expression simplifies the analysis of the series.
• This similarity suggests a deeper understanding: classic harmonic series diverge, but with our added factor \( \frac{n+\ln n}{n+10} \), different convergence patterns could emerge.

The behavior of recursive terms can consequently unveil insights about the broader characteristics and potential convergence of a series, aligning or diverging in predictable patterns.