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. $$ \left.a_{k}=\left(\frac{1}{k+1}+\frac{1}{k+2}+\cdots+\frac{1}{2 k}\right)^{k} \text { (Hint: Compare } a_{k}^{1 / k} \text { to } \int_{k}^{2 k} \frac{d t}{t} .\right) $$

Short answer

The series converges according to the root test, as \( \lim_{k \to \infty} a_k^{1/k} = \ln(2) < 1 \).

Step-by-step solution

Apply the Root Test

The root test states that for the series \( \sum a_k \), the convergence can be determined by considering \( \lim_{k \to \infty} (a_k)^{1/k} \). If this limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.

Simplify \( a_k^{1/k} \)

The given term is \( a_k = \left( \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{2k} \right)^k \). Taking the \( k \)-th root provides:\[a_k^{1/k} = \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{2k}\]because raising to the 1/k power and then to the k power cancels each other out in the exponent.

Compare to the Integral

The hint suggests comparing \( a_k^{1/k} \) to the integral \( \int_k^{2k} \frac{dt}{t} \). Evaluating this integral, we have:\[\int_k^{2k} \frac{dt}{t} = [\ln t]_k^{2k} = \ln(2k) - \ln(k) = \ln(2)\]This integral indicates that the sum of the terms in \( a_k^{1/k} \) approximates \( \ln(2) \), a known constant less than 1, for large \( k \).

Evaluate the Series Convergence

Since \( \lim_{k \to \infty} a_k^{1/k} = \ln(2) \), and because \( \ln(2) \approx 0.693 < 1 \), the root test confirms that the series \( \sum a_k \) converges.

Key concepts

Root Test

The Root Test is a powerful tool in understanding whether an infinite series converges. In simple terms, this test analyzes the behavior of the terms of the series as they approach infinity. To apply the root test to a series \( \sum a_k \), we look at the quantity \( \lim_{k \to \infty} (a_k)^{1/k} \). The basic rules are straightforward:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In our given problem, we found that \( a_k^{1/k} = \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{2k} \). After simplifying and performing the test, the limit resulted in a value less than one (specifically, it approximated to \( \ln(2) \)). Thus, by the root test, the series \( \sum a_k \) converges.

Ratio Test

The Ratio Test is another method to determine the convergence of a series and involves taking the ratio of consecutive terms. While we didn't directly use the ratio test in this problem, it's valuable to understand how it works in similar scenarios.

To apply the ratio test for a series \( \sum a_k \), calculate:

• \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \)

The conclusions from the ratio test are:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

This test is especially useful when the terms of the series involve factorials or exponential factors, making it easier to manage than some other methods. Although not used here, compare it with the root test for a deeper understanding of which scenarios one might best apply each test.

Integral Comparison

The Integral Comparison is a technique used to compare a series to a corresponding integral to determine convergence. In our exercise, it offers another perspective to assess the behavior of \( a_k^{1/k} \) and helps to confirm findings from other tests like the root test.

The idea behind this technique is to approximate a series \( \sum a_k \) through the integral \( \int f(x) \, dx \) where the terms \( a_k = f(k) \). In practice:

• Integrate over a continuous version of the series term boundaries.
• Check the integral's limit behavior to predict the series behavior.

In this problem, we evaluated the integral \( \int_k^{2k} \frac{dt}{t} \), which resulted in \( \ln(2) \). This helped us find that \( a_k^{1/k} \) approximates this value, a constant less than 1. Thus, using integral comparison alongside the root test provides a robust argument for the convergence of the series.

Convergence Tests

Convergence tests are a family of mathematical tools used to establish whether a given infinite series converges or diverges. When faced with intricate series, knowing which test to apply is crucial.

Here are several common convergence tests we can use:

• Root Test: Examines the limit of \( (a_k)^{1/k} \).
• Ratio Test: Compares successive terms using \( \frac{a_{k+1}}{a_k} \).
• Integral Test: Compares a series to an integral.
• Comparison Test: Compares the series to a known convergent or divergent series.
• Limit Comparison Test: Uses the ratio of terms with a well-known series to determine convergence.
• Alternating Series Test: For series whose terms alternate in sign.

Applying these tests correctly requires understanding the series at hand. It's about choosing the right tool to efficiently handle the series and reach a clear conclusion about its convergence.