Problem 9 - Single Variable Calculus

Question

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\)

Step-by-Step Solution

Answer

Using the Comparison Test with the harmonic series \(\sum_{n=2}^{\infty} \frac{1}{n}\) as a comparison, we find the limit of the ratio of the terms of our given series \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) and the harmonic series: \[\lim_{n\to\infty} \frac{\frac{\ln n}{n}}{\frac{1}{n}} = \lim_{n\to\infty} \ln n = \infty\] Since the limit is infinity, we conclude that the given series \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) is divergent.

Step 1: Identify a Comparing Series

In this step, we need to find a series that we can use for comparison. We are given \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\). We notice that the logarithmic function is slower in growth than the exponential function. So, we can compare our given series with a series with a simpler denominator. A good choice for comparison would be the harmonic series \(\sum_{n=2}^{\infty} \frac{1}{n}\), as it is a well-known series whose convergence properties are known (it is divergent).

Step 2: Apply the Comparison Test

Now, let's apply the Comparison Test. We want to find the relationship between the terms of our given series, \(\frac{\ln n}{n}\), and the terms of the harmonic series, \(\frac{1}{n}\). We will do this by taking the ratio of the terms of the given series to the harmonic series and find the limit as n approaches infinity: \[\lim_{n\to\infty} \frac{\frac{\ln n}{n}}{\frac{1}{n}}\]

Step 3: Simplify the Limit Expression

Now, we simplify the expression: \[\lim_{n\to\infty} \frac{\frac{\ln n}{n}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{\ln n}{n} \cdot \frac{n}{1} = \lim_{n\to\infty} \ln n\]

Step 4: Evaluate the Limit

As n approaches infinity, the value of the natural logarithm of n, \(\ln n\), also approaches infinity: \[\lim_{n\to\infty} \ln n = \infty\]

Step 5: Conclude Convergence or Divergence

Since the limit of the ratio of the terms of our given series to the terms of the harmonic series is infinity, we can conclude that our given series \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) is divergent. And there you have it! Using the Comparison Test, we have determined that the series \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) is divergent.

Key Concepts

Convergent Series

When we talk about series, one key question is whether it converges or not. A **convergent series** is a series whose sum approaches a well-defined finite number as you continue adding more terms. Imagine it as a never-ending sum that settles at some single value instead of growing indefinitely. How do you determine convergence? Well, you'll often compare your series to another known convergent series. The Comparison Test is one popular method used here.

• If a series is smaller than a known convergent series, it will also converge.
• For example, the geometric series \(\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n\) converges to 1, assuming each term is indeed halved continuously.

By knowing examples of convergent series, you can better compare and determine the properties of your unknown series using tests and logical reasoning.

Divergent Series

On the flip side, a **divergent series** is one that grows indefinitely and does not settle at a single sum. The sum becomes infinitely large as more terms are added. Generally, divergent series can either go to infinity or do not approach any particular value.
Like in our exercise, when comparing with divergent series like the harmonic series, we can conclude whether a series is divergent as well. Here's how it works with the Comparison Test:

• If your series is larger than a known divergent one, it will likely also be divergent.
• The harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) is a classic example of divergence, growing without bounds as more terms are added.

Understanding divergent series is crucial in determining that the series \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) diverges, as proven through comparison with the harmonic series.

Harmonic Series

The **harmonic series** is a famous sequence in mathematics. It takes the form \(\sum_{n=1}^{\infty} \frac{1}{n}\) and is well-known for its divergence. Despite its terms getting smaller as \(n\) increases, the overall sum keeps growing indefinitely. Let's break down why the harmonic series is significant:

• It serves as a key example of how even a series with decreasing terms can diverge.
• Due to its simple form, the harmonic series is often used in the Comparison Test to determine the divergence or convergence of other series.

In our problem, by comparing \(\sum_{n=2}^{\infty} \frac{\ln n}{n}\) with the harmonic series, we see that both show divergent behavior. This insight serves as an important learning point in understanding series and divergence.