Question

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ a_{k}=\frac{1 \cdot 4 \cdot 7 \cdots(3 k-2)}{3^{k} k !} $$

Short answer

Solution incomplete, would need further steps to be correct.

Step-by-step solution

Understand the Ratio Test

The Ratio Test can be used to determine the convergence of an infinite series where each term follows a specific pattern. For a series \( \sum a_k \), the test requires computing the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges; if \( L > 1 \), the series diverges; and if \( L = 1 \), the test is inconclusive.

Define the Terms for the Ratio

The given terms are \( a_k = \frac{1 \cdot 4 \cdot 7 \cdots (3k-2)}{3^k k!} \). We need to find \( a_{k+1} \) using the pattern of the sequence given: \( a_{k+1} = \frac{1 \cdot 4 \cdot 7 \cdots (3(k+1)-2)}{3^{k+1} (k+1)!} \). This simplifies to \( a_{k+1} = \frac{1 \cdot 4 \cdot 7 \cdots (3k+1)}{3^{k+1} (k+1)!} \).

Key concepts

Root Test

The Root Test is another powerful method used to determine whether an infinite series converges or diverges. Unlike the Ratio Test, which focuses on the ratio of successive terms, the Root Test examines the nth root of the absolute value of the nth term, denoted by \( a_n \).To apply the Root Test to a series \( \sum a_n \), compute the limit \ L = \lim_{{n \to \infty}} \sqrt[n]{|a_n|} \

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

The Root Test is especially useful when terms of the series have exponential forms. Even if both the Root and Ratio Tests offer similar insights, certain problems are more conveniently approached using this method. Remember, like other tests, positive results from the Root Test assure us about the series behavior.

Convergence Tests

Convergence tests are essential tools in determining whether an infinite series converges or diverges. When dealing with sequences and series, knowing what test to apply is crucial.Some common convergence tests include:

• Ratio Test: Compares successive terms of a series and focuses on their limit.
• Root Test: Utilizes the nth root approach to decide on convergence.
• P-Series Test: Useful when dealing with series in the form of \( \frac{1}{n^p} \).
• Integral Test: Involves comparing a series to a related integral for convergence.

When using these tests, it is crucial to understand that they are not always definitive, especially in cases where results are inconclusive. In such scenarios, trying a different type of test or reformulating the series might help. Each test has its strength, depending on the form and nature of the series in question.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It is represented mathematically as \( \sum_{k=0}^{\infty} a_k \), where \( a_k \) are the terms of the sequence. Understanding infinite series is foundational in calculus and mathematical analysis.An infinite series can either converge or diverge:

• Convergent: If the partial sums of the series approach a finite limit as the number of terms increases.
• Divergent: If the partial sums do not approach a finite limit.

When working with infinite series, especially with large or complex terms, applying tests like the Ratio or Root Test aids in ascertaining their convergence status. For example, if each term \( a_k = \frac{1 \cdot 4 \cdot 7 \cdots (3k-2)}{3^k k!} \) follows a determinate pattern, evaluating its convergence becomes methodical and approachable.

Sequences

A sequence is a set of numbers listed in a specific order, where each number is called a term. Sequences can be finite or infinite. Understanding sequences is crucial for exploring series as they form the building blocks of these complex mathematical structures. Characteristics of sequences:

• Finite Sequence: Has a last term.
• Infinite Sequence: Continues indefinitely.
• Arithmetic Sequence: The difference between consecutive terms is constant.
• Geometric Sequence: Each term is a multiple of the previous one by a constant factor.

The behavior of sequences affects the sequence’s corresponding series. For example, a rapidly growing sequence can indicate divergence in its series. Conversely, a sequence where terms decrease towards zero suggests potential convergence. Understanding both sequences and their derived series aids in performing convergence tests effectively.