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. $$ a_{k}=\left(\frac{1}{k+1}+\frac{1}{k+2}+\cdots+\frac{1}{3 k}\right)^{k} $$

Short answer

The series diverges by the root test.

Step-by-step solution

Analyze the Series

We observe the series given by the terms \(a_k = \left(\frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{3k}\right)^k\). We need to determine the convergence of this series using either the ratio test or the root test.

Choose an Appropriate Test

For series that involve terms raised to the power of \(k\), the root test is often a suitable choice. Therefore, we decide to use the root test to determine the convergence of this series.

Apply the Root Test

The root test states that we need to calculate \( \lim_{k \to \infty} \sqrt[k]{a_k} \). If the result is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Simplify the Expression Inside the Limit

Calculate the expression \( \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{3k} \). This is a sum of harmonic terms. Approximating, this is close to the integral from \(k+1\) to \(3k\), giving roughly \(\ln(3) - \ln(1) = \ln(3)\). This approximation helps simplify the calculations.

Calculate the Limit Using the Root Test

With \(a_k = \left(H_k\right)^k\), where \(H_k\) is our approximation of the harmonic series, compute: \[ \lim_{k \to \infty} \sqrt[k]{a_k} = \lim_{k \to \infty} \left( \ln 3 \right) = \ln(3) \].Since \(\ln(3) > 1\), the root test indicates divergence.

Key concepts

Ratio Test

The ratio test is a useful convergence test for handling series with terms that involve complex expressions or involving factorials, exponentials, or other rapid growth.

• The key idea is to examine the ratio of successive terms in the series.
• For a series \( \sum a_k \), compute the limit \( \lim_{k\to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
• If this value is less than 1, the series converges.
• If the value is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive, and another method is needed.

The ratio test is especially effective for series like geometric series or those that behave similarly to geometric series. It provides a quick and often straightforward way to check for convergence or divergence, saving time over more complex methods.

Root Test

The root test, also known as the Cauchy root test, is especially helpful for series whose terms involve powers. This is particularly true when a term in the sequence includes forms like \(a_k^k\).

• To apply this test, calculate \( \lim_{k \to \infty} \sqrt[k]{\left| a_k \right|} \).
• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

Choosing the root test for the given series was appropriate because terms are raised to the power of \(k\). The root test efficiently handles such expressions, providing clarity in determining convergence or divergence through direct computation of limits.

Harmonic Series

The harmonic series is the simplest example of a divergent series despite having terms that decrease to zero. This series is represented as \( \sum_{k=1}^{\infty} \frac{1}{k} \).

• Despite its terms decreasing, its partial sums grow without bound, hence it diverges.
• The harmonic series often appears in various mathematical problems and can be related to natural logarithms.
• Summing terms from a harmonic sequence results in a logarithmic relationship, which is useful for approximation.
• In step 4 of the original solution, the approximation of the sum of terms using integrals leverages the logarithmic nature of the harmonic series.

Understanding the properties of the harmonic series can be critical in solving problems where harmonic-like sequences appear, whether they stand alone or are part of more complex series. It highlights the importance of careful approximation in convergence tests.