Problem 15 - Thomas Calculus

Question

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=2}^{\infty} \frac{1}{\ln n}} \\\ {\text {(Hint: Limit Comparison with } \sum_{n=2}^{\infty}(1 / n) )}\end{array}\end{equation}

Step-by-Step Solution

Answer

The series \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\) diverges.

Step 1: Identify the Series

We need to analyze the series \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\) and determine its convergence or divergence.

Step 2: Choose a Comparison Series

According to the hint, we will use the series \(\sum_{n=2}^{\infty} \frac{1}{n}\) for comparison, which is a well-known divergent series.

Step 3: Apply the Limit Comparison Test Formula

The Limit Comparison Test involves calculating \( c = \lim_{n \to \infty} \frac{a_n}{b_n} \) where \( a_n = \frac{1}{\ln n} \) and \( b_n = \frac{1}{n} \).

Step 4: Calculate the Limit

We find:\[ c = \lim_{n \to \infty} \frac{\frac{1}{\ln n}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{\ln n}.\]To evaluate this limit, recognize that as \( n \to \infty \), \( \ln n \to \infty \) and grows slower than \( n \). Therefore, \( \frac{n}{\ln n} \to \infty \).

Step 5: Interpret the Limit Comparison Test Result

Since the limit \( c = \infty \) (non-zero and positive), according to the Limit Comparison Test, the behavior of the series \( \sum_{n=2}^{\infty} \frac{1}{\ln n} \) is the same as the series \( \sum_{n=2}^{\infty} \frac{1}{n} \), which diverges.

Key Concepts

Convergent and Divergent Series

When exploring series in mathematics, it's important to understand the concepts of convergence and divergence. This helps us determine whether an infinite series adds up to a finite number or not. A series is called *convergent* if the sum of its terms approaches a specific number as you keep adding more terms. In simpler terms, if you could add up all the numbers in a convergent series, you would reach a specific and finite total.

• If a series approaches a finite limit as more terms are included, it's convergent.
• If a series does not approach any limit and keeps increasing or oscillating indefinitely, it's divergent.

On the other hand, a *divergent* series is one where the sum of its terms grows indefinitely or does not settle on a finite limit. In the exercise, we came across the series \( \sum_{n=2}^{\infty} \frac{1}{\ln n} \), which was determined to *diverge* using the Limit Comparison Test. The \(\sum_{n=2}^\infty \frac{1}{n}\) series, which is known to diverge, was used in the comparison process to establish this result.

Infinite Series

An infinite series is a sum that includes an infinite number of terms. This might seem daunting at first, but mathematicians have developed several methods to analyze these series.
These methods help us determine whether their sums can be calculated meaningfully.

• The *nature* of an infinite series depends significantly on its terms. If the terms decrease rapidly, the series may converge.
• If the terms do not decrease fast enough, the series likely diverges.

The given exercise involved analyzing an infinite series \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\), which challenges us to consider how these terms decrease as \(n\) becomes larger. While each term becomes smaller as \(n\) goes to infinity, the challenge is to determine if they decrease fast enough for the series to sum up to a finite number. Through comparison with the harmonic series, which is known to diverge, it was shown that the given series must also diverge.

Comparison Tests for Series

Comparison tests are powerful tools in determining the convergence or divergence of a series. They involve comparing a series of interest to a second series whose convergence properties are already known and understood.
There are mainly two types of comparison tests: the Direct Comparison Test and the Limit Comparison Test.

Direct Comparison Test

The Direct Comparison Test involves directly comparing each term of one series to the corresponding term of another series. If the series we are investigating has terms smaller than a convergent series, it must also converge. On the flip side, if it has terms larger than a known divergent series, it will diverge.

Limit Comparison Test

The Limit Comparison Test, which was used in our exercise, calculates the limit of the ratio of the terms of the two series as \(n\) approaches infinity. In our case, the exercise involved the infinite series \(\sum_{n=2}^{\infty} \frac{1}{\ln n}\) compared to \(\sum_{n=2}^\infty \frac{1}{n}\).

• If the limit is a positive finite number, both series will either converge or diverge together.
• If the limit is zero or infinity, the test is inconclusive, and another method must be used.

The exercise showed that the limit was infinity, meaning both series diverge. Understanding how these tests work and when to apply them is crucial for analyzing any given series efficiently.