Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=1}^{\infty}(-1)^{k+1} \frac{k !}{(k+1) !} $$

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k !}{(k+1) !}\) converges.

Step 1: Identify the Series

The series provided is an alternating series because of the term \((-1)^{k+1}\). This causes the sign to alternate between positive and negative. Let's denote the series as \(\sum_{k=1}^{\infty} a_k\), where \(a_k = (-1)^{k+1} \frac{k !}{(k+1) !}\).

Step 2: Simplify the Expression

The expression \(a_k = (-1)^{k+1} \frac{k !}{(k+1) !}\) can be simplified. We know that \((k+1)! = (k+1) \times k!\). Using this, our expression becomes \(a_k = (-1)^{k+1} \frac{1}{k+1}\).

Step 3: Apply the Alternating Series Test

The Alternating Series Test states that a series \(\sum (-1)^n a_n\), where all \(a_n\) are positive, converges if the sequence \(\{a_n\}\) is decreasing and \(\lim_{n\to\infty} a_n = 0\). Here, our series meets these conditions because \(a_k = \frac{1}{k+1}\) is decreasing and \(\lim_{k\to\infty} a_k = 0\). Therefore, the series converges.

Key Concepts

Alternating Series

An alternating series is a series where the terms change sign in a regular pattern. In this exercise, the series takes the form

• \((-1)^{k+1}\),

which simply means you alternate between adding and subtracting successive terms. This pattern is characterized by multiplying each term by either \(1\) or \(-1\). Understanding this sign change is crucial for analyzing the series.
Alternating series are important in mathematics because they often converge even when similar non-alternating series do not. They have unique properties that can be exploited to determine convergence or divergence.
To study convergence, observe the behavior of the absolute values of these terms over time, as they tend to decrease in magnitude.

Factorial Simplification

Factorial simplification is key to handling complex series and easing calculations. The factorial of a number \(n\), denoted as \(n!\), is the product of all positive integers up to \(n\). Therefore, \((k+1)! = (k+1) \times k!\).
In our series expression, \(\frac{k!}{(k+1)!}\), we can simplify using this identity. By replacing \((k+1)!\) with \((k+1) \times k!\), the terms \(k!\) cancel each other, simplifying to

• \(\frac{1}{k+1}\).

Factorial simplification helps reduce complex expressions, making it easier to analyze their properties, including convergence.

Alternating Series Test

The Alternating Series Test is a vital tool for determining the convergence of alternating series. To utilize this test, you need to check two specific conditions:

• The sequence of terms \(a_n\) is positive and decreases steadily as \(n\) increases.
• The limit of the sequence \(a_n\) as \(n\) approaches infinity is zero, i.e., \(\lim_{n \to \infty} a_n = 0\).

Applying these conditions to our series, we see that:

• The terms \(\frac{1}{k+1}\) get smaller as \(k\) becomes larger, fulfilling the first condition.
• Further, the limit \(\lim_{k \to \infty} \frac{1}{k+1} = 0\), satisfies the second condition.

Thus, according to the test, our series converges. The alternating series test is a straightforward method to conclude if an alternating series converges, regardless of complexity.