Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty} \frac{1}{\ln n}$$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \) diverges by the Integral Test.

Step-by-step solution

Check if Series Meets Basic Requirements

The given series is \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \). We first check to see if this series is a p-series or an easy comparison series but quickly find that it's neither because of the \(\ln n \) denominator.

Apply the Integral Test

To determine the convergence, apply the Integral Test. To use the Integral Test, consider \( f(x) = \frac{1}{\ln x} \) which is continuous, positive, and decreasing for \( x > 1 \). Evaluate the integral \( \int_{2}^{\infty} \frac{1}{\ln x} \, dx \).

Solve the Integral

The integral \( \int \frac{1}{\ln x} \, dx \) can be solved by substituting \( u = \ln x \), hence \( du = \frac{1}{x} \, dx \). The integral becomes \( \int \frac{1}{u} \, du = \ln |u| + C = \ln |\ln x| + C \). Evaluate the improper integral \( \int_{2}^{\infty} \frac{1}{\ln x} \, dx \) by substituting back.

Evaluate the Improper Integral

Now verify the convergence or divergence of \( \lim_{t \to \infty} \ln |\ln t| - \ln |\ln 2| \). As \( t \to \infty \), \( \ln(\ln t) \to \infty \), implying the integral diverges.

Conclude Divergence of the Series

Since the integral diverges, by the Integral Test, the original series \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \) also diverges.

Key concepts

Convergence

Convergence is a concept that deals with whether a series approaches a finite limit as the number of terms increases to infinity. A series is said to be convergent if the sum of its terms approaches a specific value, rather than increasing or decreasing without bound. To understand convergence better, let's break down the conditions necessary for it:

• The series must be made up of terms from a sequence that becomes smaller and smaller as you add more terms.
• Adding more terms doesn't change the total sum significantly after a certain point.

For the series \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \), the question is whether its infinite sum truly converges to a limit. Some typical tests for convergence include the Integral Test, p-Series Test, and Alternating Series Test.
Regarding this exercise, applying the Integral Test revealed that the series does not converge because the integral evaluated from 2 to infinity tends to infinity. This tells us that the sum of the series diverges instead of converging. More on divergence next.

Divergence

In contrast to convergence, divergence occurs when the sum of an infinite series keeps growing larger or smaller without approaching a specific limit. For a series to be divergent, its terms do not settle towards zero sufficiently fast, or when summed, they fail to reach a finite sum.

• Series diverge if their terms do not approach zero as \( n \to \infty \).
• Some series might oscillate or increase indefinitely.

In our exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \) was found to be divergent by applying the Integral Test. The key factor was that the integral \( \int_{2}^{\infty} \frac{1}{\ln x} \, dx \) went to infinity, implying that the series itself also grows indefinitely. If an integral related to a series fails to converge, this is a strong indication that the series itself is divergent.

Series

A series is the sum of the terms of a sequence. It's an infinite process of adding up all the numbers in a specific order, usually given in a sequence, to explore and understand their collective value. The notation \( \sum_{n=1}^{\infty} a_n \) represents an infinite series, which can either converge or diverge based on the behavior of the sequence \( a_n \).

• There's a distinction between finite series (those with a limited number of terms) and infinite series.
• Infinite series, like in our problem, extend indefinitely, and their behavior (converging or diverging) is a key concept in calculus.

In this specific exercise, the series \( \sum_{n=1}^{\infty} \frac{1}{\ln n} \) is infinite because it sums up terms that extend to infinity as \( n \) increases. Various tests, like the Integral Test, can determine whether these series yield a finite sum or not. Understanding the nature of a series is crucial for correctly applying these tests and understanding their results in mathematical exercises.