Problem 4 - Thomas Calculus

Question

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{4}{(\ln n)^{2}} $$

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Check if Terms Decrease

To use the alternating series test, we must first verify that the absolute value of the terms decreases as \( n \) increases. Consider the terms \( b_n = \frac{4}{(\ln n)^2} \). Determine if \( b_n > b_{n+1} \) for sufficiently large \( n \). Calculate \( b_{n+1} \) and check if this condition holds.

Step 2: Calculate Limiting Behavior of Terms

Next, we need to ensure that the limit of the terms \( b_n \) as \( n \) approaches infinity is zero. Calculate \( \lim_{n \to \infty} \frac{4}{(\ln n)^2} \). As \( n \) becomes very large, \( \ln n \) also increases, causing \( \frac{4}{(\ln n)^2} \) to approach zero.

Step 3: Apply Alternating Series Test

The Alternating Series Test states that an alternating series \( \sum_{n=1}^{\infty} (-1)^n b_n \) converges if the absolute value of its terms \( b_n \) decreases monotonically and \( \lim_{n \to \infty} b_n = 0 \). Since both conditions are satisfied (from Steps 1 and 2), apply this test to conclude that the series converges.

Key Concepts

Convergence of series

A series, in mathematics, is the sum of the terms of a sequence. One crucial question about a series is whether it converges to a specific value or diverges. The convergence of a series depends on the behavior of its terms as the series progresses to infinity.

**How to Determine Convergence?**

• If the terms of the series do not tend to zero, the series can't converge.
• Additional tests like the Alternating Series Test are often used to determine convergence, especially for alternating series.

A convergent series has a finite sum, which means the partial sums of the series approach a specific finite number as more terms are added.

In our example, we use the Alternating Series Test to check the convergence of the given series by verifying certain conditions. If these conditions are satisfied, the series converges, providing a neat method to judge complex series.

Alternating series

An alternating series is a series where the terms alternate in sign, meaning a positive term is followed by a negative one, and vice versa. This pattern can significantly influence the convergence of the series.

**Characteristics of Alternating Series:**

• Terms switch between positive and negative.
• Often denoted as \( \sum (-1)^n a_n \), where the sequence \( a_n \) includes only positive terms.
• Very useful for series where the standard tests may not straightforwardly apply.

For the Alternating Series Test, two main criteria need verification:
1. The absolute value of terms must decrease monotonically, which means they shrink consistently from one term to the next.2. The terms must approach zero as the series extends towards infinity.
If both conditions are satisfied, as in our exercise, the series converges. This test offers a direct way to conquer the often-complex behavior of alternating series.

Limit of a sequence

The limit of a sequence is key to understanding series behavior. It describes what value, or set of values, the terms of the sequence approach as the sequence progresses indefinitely.

**Understanding Limits with Examples:**

• If the smallest term in the sequence drops to zero, it signals possible convergence of the associated series.
• For mathematical rigor, \( \lim_{n \rightarrow \infty} a_n = L \) is used to express the concept.

In the context of our series, we examine \( \frac{4}{(\ln n)^2} \). As \( n \) goes to infinity, \( \ln n \) grows slowly but surely, driving the fraction towards zero. When a term of a series satisfies this limit property, it serves as an indicator—together with other conditions from tests like the Alternating Series Test—that the series might converge.

When the limit of the sequence terms equals zero, the groundwork is solid for the series to potentially add up to a finite value, indicating convergence.