Problem 95 - 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. $$ a_{1}=\frac{1}{3}, a_{n+1}=\left(1+\frac{1}{n}\right) a_{n} $$

Step-by-Step Solution

Answer

The convergence or divergence of the series can't be determined from the information given and the tests used.

Step 1: Apply the Ratio Test

For the Ratio Test, we need to calculate the limit of the absolute ratio of two consecutive terms, \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). For the given series, this is \(L = \lim_{n \to \infty} (1+ \frac{1}{n})\).

Step 2: Evaluate the Limit

The term inside the limit tends towards 1 as \(n \to \infty\) because the term \(\frac{1}{n}\) tends towards 0. Therefore, \(L = 1\).

Step 3: Conclude Based on the Ratio Test

According to the Ratio Test, if \(L < 1\), the series converges, if \(L > 1\), it diverges, and if \(L = 1\), the test is inconclusive. Hence, in this case as \(L = 1\), the Ratio Test doesn't provide enough information for us to determine the convergence or divergence of the series.

Step 4: Use Other Tests

Since the Ratio Test was inconclusive, we should attempt other tests. However, the information provided doesn't give us a direct way to use other popular tests like the Root Test or the Comparison Test. Therefore, the question can't be resolved with the information given.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool for determining the convergence of infinite series, particularly where terms are defined by a relation between consecutive terms. The beauty of the Ratio Test lies in its simplicity: it involves calculating the limit of the ratio between consecutive terms.

When applying the Ratio Test, we compute the ratio of the absolute value of the terms, \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). The series will converge if \( L < 1 \) and diverge if \( L > 1 \). If the limit equals 1, however, the test is inconclusive. In our exercise, where the terms of the series grow according to a recursive formula, the Ratio Test initially seemed promising. Upon finding \( L = 1 \), however, it could not help us ascertain whether the series converges or diverges.

Recursive Series

Recursive series are sequences of numbers where each term is defined based on the preceding one. In essence, they are the numerical version of a family tree, each member (term) is born from the previous generation (term).

A classic example is our exercise's \( a_{n+1} = \left(1 + \frac{1}{n}\right)a_{n} \), where each term grows out from the previous one like branches on a tree. While recursive series can form beautiful and complex mathematical structures, they can also pose a significant challenge when it comes to determining their convergence. Our core question keeps revolving around whether these sequential generations will eventually settle down (converge) or keep growing outwards endlessly (diverge).

Limit of a Sequence

Imagine a child on a swing, each swing reaching a little less high, until eventually, the movement settles into stillness. This is the concept of a limit in a nutshell — it describes where a sequence heads as the terms progress, even if it never actually gets there.

In mathematical language, we express this as \( \lim_{n \to \infty} a_n \). To grasp whether a series converges, we often look at the limit of its termed sequence as \( n \) approaches infinity. However, determining the simple existence of this limit is not always synonymous with convergence of the series, as we learned in the given exercise.

Convergence Tests

Like a doctor’s toolkit, convergence tests are a set of diagnostic tools mathematicians use to understand the behavior of series. Each test has its own strengths and particular series where it applies best.

While the Ratio Test is excellent for series with factorials or exponential growth, other tests, like the Root Test or the Comparison Test, might be better suited for different series patterns. As highlighted in step 4 of the problem solution, when one test fails to provide clarity, it's time to probe the series with other tests. Unfortunately, without additional information or more terms of the series, we are left with an unresolved diagnosis. Distinguishing between tools is essential, as selecting the right test can be as crucial as the test's outcome itself.