Problem 94 - Essential Calculus

Question

The terms of a series \(\sum_{n=1}^{\infty} a_{n}\) are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. $$ a_{1}=\frac{1}{5}, a_{n+1}=\frac{\cos n+1}{n} a_{n} $$

Step-by-Step Solution

Answer

The convergence or divergence of the series \(\sum_{n=1}^{\infty} a_{n}\) can be determined by using the divergent test. If the limit of the general term \(a_n\) as \(n\) approaches infinity is nonzero, then the series is divergent.

Step 1: Understanding the recursive relation

Given how \( a_{n+1} \) relates to the previous term, \( a_n \), and the index, \( n \), write down the first few terms to develop an understanding. If \( a_{1} = \frac{1}{5} \) then: \( a_{2} = \frac{\cos(1) + 1}{1} * \frac{1}{5} \), \( a_{3} = \frac{\cos(2) + 1}{2} * a_{2} \), and so on.

Step 2: Observing the behavior of the sequence

Use the recursive relation to calculate more terms of the sequence \( a_n \). This will help identify a pattern and make a hypothesis about convergence or divergence.

Step 3: Test for divergence

Applying the test for divergence, if the limit of a general term, \( a_n \), as \( n \) approaches infinity is not equal to zero, then the series \(\sum_{n=1}^{\infty} a_{n}\) is divergent. Calculating \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} (\frac{\cos n + 1}{n})*a_{n-1} \), if this is not equal to zero, the given series is divergent.

Key Concepts

Recursive Sequences

In mathematics, a sequence is called recursive if each term is defined as a function of the preceding terms. The given problem presents a recursive sequence where each term after the first is calculated by a defined rule involving the previous term. For example, in our exercise, each subsequent term is derived by multiplying the previous term by a fraction whose numerator is \( \cos(n) + 1 \) and the denominator is \( n \) itself.

To visualize a recursive sequence, it's helpful to start by computing a few initial terms. By plugging in the values of \( n \) into our recursive formula, we observe how the sequence evolves. This hands-on approach can illuminate patterns that suggest behaviors like convergence or divergence, which are fundamental to understanding the sequence's long-term tendencies.

Test for Divergence

Determining the fate of a sequence or series—whether it 'settles down' to a limit or sprawls outward indefinitely—is critical in analysis. The test for divergence is a simple yet powerful tool for series analysis. It states that if the limit of the sequence's terms \( a_n \) does not approach zero as \( n \) goes to infinity, then the series \( \sum_{n=1}^{ty} a_{n} \) diverges.

For the given exercise, applying this test involves finding the limit of \( a_n \) as \( n \) becomes very large. If this limit is not zero, the whole series cannot possibly converge to a finite value. That is, if the individual terms of the series don't become negligible, their sum cannot stabilize. Hence, this quick test can immediately confirm the divergence of a series without a need for more complex analysis.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Specifically, if we have a sequence \( {a_n} \) where \( n \) takes on values from 1 to infinity, the sum \( \sum_{n=1}^{\infty} a_{n} \) represents an infinite series. These series can sometimes add up to a finite value, which we call the series' sum, or they might grow without bound.

Understanding whether the series converges (sums to a finite number) or diverges (increases indefinitely) is fundamental in mathematics. Problems in this domain often involve testing a series for convergence using various techniques, such as comparison tests, ratio tests, the root test, and the aforementioned test for divergence.

Limits of Sequences

The limit of a sequence is the value that the sequence's terms approach as the index \( n \) becomes very large. To find the limit, we look for a pattern or a behavior that the terms follow as \( n \) grows without bound and determine what value the terms get 'closer' to. In our exercise's context, we're specifically interested in the behavior as \( n \) approaches infinity.

Mathematically, we denote this by \( \lim_{n \to \infty} a_n \). If this limit exists and is a finite number, the sequence is said to converge to that number. If the limit doesn’t exist or is infinite, the sequence is said to diverge. The limit concept is not just foundational for sequences but for all of calculus, as it formalizes the notion of approaching a particular value without necessarily ever reaching it.