Problem 8 - 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. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{10^{1}}{(n+1) !} $$

Step-by-Step Solution

Answer

The alternating series converges.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty}(-1)^{n} \frac{10}{(n+1) !} \). It is an alternating series due to the factor \((-1)^n\), as the terms alternate in sign.

Step 2: Confirm Conditions for Alternating Series Test

The Alternating Series Test requires that for the series \( \sum (-1)^n a_n \), the sequence \( a_n \) meets two conditions: (1) \( a_n \) must be positive, and (2) \( a_n \) must decrease to zero as \( n \to \infty \). Here, \( a_n = \frac{10}{(n+1)!} \) is positive for all \( n \geq 1 \) and decreases because factorials grow rapidly.

Step 3: Check if Terms Decrease to Zero

Examine whether \( \lim_{n \to \infty} a_n = 0 \). Since \( a_n = \frac{10}{(n+1)!} \) and factorials grow much faster than any constant or polynomial function, \( \lim_{n \to \infty} \frac{10}{(n+1)!} = 0 \). This satisfies the second condition of the test.

Step 4: Conclusion on Convergence

Since both conditions of the Alternating Series Test are satisfied (\( a_n > 0 \) and \( a_n \to 0 \)), the series \( \sum_{n=1}^{\infty}(-1)^{n} \frac{10}{(n+1)!} \) converges.

Key Concepts

Convergence of Series

In mathematics, a series is said to "converge" if the sum of its terms approaches a particular value as the number of terms increases to infinity. Convergence is a fundamental concept when analyzing series. For alternating series, the Alternating Series Test is used to determine convergence. This test checks two main things:

• The terms of the series, often denoted as \(a_n\), must be positive. That means, each \(a_n\) should be greater than zero.
• The sequence of terms \(a_n\) must decrease to zero as \(n\) approaches infinity.

In practice, if both conditions are met, the series is said to be convergent. Conversely, if even one condition is not satisfied, the series may diverge, meaning it does not sum to a finite number. In our example, since the series satisfies both conditions, it is convergent.

Factorials

Factorials, denoted with an exclamation mark (like \(n!\)), are the product of all positive integers up to a certain number \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow extremely quickly as the value of \(n\) increases.
This rapid growth is crucial in the analysis of series involving factorials.

• Because factorials increase quickly, fractions with factorials in the denominator can tend to zero. For instance, \(\frac{1}{n!}\) decreases very swiftly as \(n\) becomes larger.
• Using factorials in series analysis can demonstrate how terms decrease to zero, satisfying a key condition of the convergence tests, like the Alternating Series Test.

Therefore, understanding how factorials behave helps to predict whether series involving them will converge.

Series Analysis

Analyzing a series involves determining whether it converges or diverges and understanding the behavior of its terms. Key steps often include identifying the type of series and applying appropriate convergence tests.
In this context, alternating series have distinctive characteristics:

• They contain terms that alternate in sign. This is typically represented as \((-1)^n\) or \((-1)^{n+1}\) in each term, causing the terms to switch from positive to negative or vice versa.
• Checking decrease to zero and positivity are essential steps as illustrated by the Alternating Series Test. Knowing how the growth of terms affects the series is vital.

A deeper understanding of terms and dynamics, like gradually decreasing to zero or role of factorials, can guide predictions about convergence. Step-by-step analysis, as performed in the exercise, allows us to make informed conclusions about series behaviors.