Chapter 1: Problem 5

Test the following series for convergence. \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}\)

Short Answer

Expert verified

The series converges by the Alternating Series Test.

Step by step solution

Title - Identify the Type of Series

The given series is \(\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}\). This is an alternating series because of the \((-1)^{n}\) factor.

Title - Determine if the Alternating Series Test (Leibniz Test) Applies

To use the Alternating Series Test, confirm the following conditions: \(a_n = \frac{1}{\ln n}\) is positive, decreasing, and \(\lim_{n \to \infty} a_n = 0\).

Title - Verify Positivity

For \(n \geq 2\), the logarithm function \(\ln n\) is positive, thus \(a_n = \frac{1}{\ln n} > 0\).

Title - Verify Decreasing Nature

Consider the derivative of \(a_n = \frac{1}{\ln n}\). If the derivative is negative for \(n \geq 2\), then \(a_n\) is decreasing. Using the derivative of \(\frac{1}{u}\) where \(u = \ln n\): \(\frac{d}{dn} \left( \frac{1}{\ln n} \right) = -\frac{1}{n (\ln n)^2}\). Since \(-\frac{1}{n (\ln n)^2} < 0\) for \(n \geq 2\), \(a_n\) is decreasing.

Title - Verify the Limit Condition

Evaluate \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{\ln n}\). As \(n\) approaches infinity, \(\ln n\) also approaches infinity, making \(\frac{1}{\ln n}\) approach 0. Hence, \(\lim_{n \to \infty} \frac{1}{\ln n} = 0\).

Title - Conclusion by Alternating Series Test

Since all three conditions of the Alternating Series Test are satisfied \((a_n > 0\), \(a_n\) is decreasing, and \(\lim_{n \to \infty} a_n = 0\)), the given alternating series converges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Alternating Series Test

An alternating series is a series in which the terms alternate in sign. You might notice it because it has a \((-1)^n\) or \((-1)^{n+1}\) factor. A common test to determine the convergence of an alternating series is the Alternating Series Test, also known as the Leibniz Test. This test ensures a series converges if three conditions are met:

The terms \(a_n\) (ignoring the sign) must be positive.
The terms \(a_n\) should be decreasing.
The limit of \(a_n\) as \(n\) approaches infinity should be zero.

To put it simply, check if the positive terms get smaller and tend towards zero. If they do, the series converges.

Logarithmic Function

The logarithmic function, denoted as \(\text{ln} n\), is crucial in calculus and real analysis. It gives the power to which a base (usually the number \(e \approx 2.718\)) must be raised to produce a given number. For example, the logarithm of \(e\) itself is \(1\) because \(e^1 = e\). Some properties to keep in mind about logarithms:

\(\text{ln}(mn) = \text{ln}(m) + \text{ln}(n)\)
\(\text{ln}(m/n) = \text{ln}(m) - \text{ln}(n)\)
\(\text{ln}(a^b) = b \text{ln}(a)\)

In this exercise, calculating \( a_n = \frac{1}{\text{ln} n} \) helps us understand the terms of the series.

Decreasing Sequence

In mathematics, a sequence is decreasing if each term is less than or equal to the previous term. This means with each step, the value of the term gets smaller or stays the same. For a sequence \(a_n\), if we can show that \(a_{n+1} \leq a_n \), it is decreasing. To prove this formally, we often use the derivative:

Find the derivative of the term.
If the derivative is negative for all \(n \geq 2\), it means the original term is decreasing.

In our series, by showing that the derivative of \( \frac{1}{\text{ln} n} \) is negative, we proved the terms decrease.

Series Convergence

A series converges if the sequence of its partial sums approaches a finite limit. That is, as we add more terms, the sum eventually stabilizes at a certain value. Different tests can determine the convergence of a series, such as the Alternating Series Test, Ratio Test, and Root Test. Convergence is key in understanding whether adding infinitely many terms will give us a meaningful result. In the given series, we confirmed convergence using the Alternating Series Test, demonstrating the sum's stability as an infinite series.

Mathematical Proof

In mathematics, a proof is a logical argument demonstrating the truth of a proposition. Proofs are built from axioms, definitions, and previously established theorems. Key steps usually involve:

Identifying what you need to prove.
Using known theorems and rules to build your argument.
Arriving at a conclusion that follows logically from your steps.

In the solution above, we used a structured approach: defining the series, applying the Alternating Series Test, and thus proving the convergence of our series logically and systematically. This methodical process ensures the proof is both correct and understandable.