Problem 27 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=2}^{\infty} \frac{\sqrt{n}}{\ln n} $$

Step-by-Step Solution

Answer

The series diverges by the Limit Comparison Test with \( \sum \frac{1}{n^{1/2}} \).

Step 1: Identify the Series

The given series is \( \sum_{n=2}^{\infty} \frac{\sqrt{n}}{\ln n} \). We need to determine whether this series converges or diverges.

Step 2: Apply the Limit Comparison Test

To use the Limit Comparison Test, choose a comparison series that closely resembles the given series but has a known behavior. Compare \( \frac{\sqrt{n}}{\ln n} \) to \( \frac{1}{n^{1/2}} \), which is the harmonic series raised to the power 1/2 and is known to diverge.

Step 3: Calculate the Limit for Comparison

Calculate \( \lim_{n \to \infty} \frac{\frac{\sqrt{n}}{\ln n}}{\frac{1}{n^{1/2}}} = \lim_{n \to \infty} \frac{n}{\ln n} \). As \( n \to \infty \), \( \frac{n}{\ln n} \to \infty \), implying that both series grow at the same rate.

Step 4: Determine Convergence or Divergence

Since \( \lim_{n \to \infty} \frac{n}{\ln n} \to \infty \) and this ratio is positive, the given series \( \sum_{n=2}^{\infty} \frac{\sqrt{n}}{\ln n} \) has the same divergence behavior as the divergent series \( \sum_{n=2}^{\infty} \frac{1}{n^{1/2}} \). Thus, by the Limit Comparison Test, the original series diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool that helps us answer a crucial question: does a given series converge or diverge? It's especially handy when direct examination of a series doesn't make the convergence behavior obvious. The test works by comparing the series in question with another series, whose convergence properties are already known.

**How it works:**

• Select a comparison series. This series should be similar to the original series in terms of its general form but should have a known convergence or divergence status.
• Compute the limit as follows: For a given series \( \sum a_n \) and a comparison series \( \sum b_n \), calculate \( \lim_{n \to \infty} \frac{a_n}{b_n} \).

If this limit is a finite positive number (i.e., greater than zero but less than infinity), then both series either converge or diverge. If \( \sum b_n \) diverges, like in our exercise, then \( \sum a_n \) also diverges. It's crucial that the selected comparison series matches the complexity of the series under consideration. A solid choice of comparison can simplify analysis dramatically.

By humanizing this process with a step-by-step approach, the Limit Comparison Test becomes an extremely practical method, especially for students working through challenging series on their own.

Divergent Series

A series is considered divergent when the sequence of partial sums doesn't settle towards a fixed value. In other words, the sum grows indefinitely. Divergent series appear quite frequently in mathematical analysis, and recognizing them is vital for understanding the behavior of infinite sums.

**Key Characteristics of a Divergent Series:**

• The partial sums do not approach a specific numerical limit.
• When tested, phenomena like the harmonic series show that their positive sequence terms continue to add increasingly without bound.

Divergence in a series often signals an infinite accumulation, echoing concepts of growth in natural or theoretical processes. For instance, in the given exercise, our series was compared to \( \sum_{n=2}^{\infty} \frac{1}{n^{1/2}} \), a well-known divergent series due to its resemblance to a harmonic series. Spotting such characteristics early on makes the task of classifying series simpler. Beware, though: not all divergent series are easy to pinpoint, and often require thorough testing using various convergence tests.

Convergence Tests

Convergence tests are methods used to determine whether a series converges or diverges. Different series require different tests, depending on their structure and form. These tests are indispensable tools in mathematical analysis and calculus.

**Popular Convergence Tests:**

• Ratio Test: useful for series whose terms involve factorials or power functions. It examines the limit of the ratio of successive terms.
• Root Test: similar to the Ratio Test but uses the \( n \)-th root of terms, often handy when terms involve exponents.
• Integral Test: applicable when terms are positive and decreasing, integrating terms as if they were part of a function.
• Limit Comparison Test: as described, this contrasts terms with a benchmark series.

Each test has its specialty, and knowing when to apply which one is fundamental. In our exercise, the Limit Comparison Test was ideal due to the presence of a complicated logarithmic function, along with a root function. By comparing against a known divergent series, a clear result was achieved. Selecting the right convergence test can simplify the process of determining the nature of a series, making it essential to mastering calculus.