Problem 3 - Thomas Calculus

Question

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{(n-1) !}{(n+1)^{2}}$$

Step-by-Step Solution

Answer

The series diverges by the Ratio Test.

Step 1: Write Down the Series

The given series is \( \sum_{n=1}^{\infty} \frac{(n-1)!}{(n+1)^{2}} \). Our task is to determine if this series converges absolutely using the Ratio Test.

Step 2: Write the Formula for the Ratio Test

The Ratio Test involves the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n \) is the general term of the series, \( a_n = \frac{(n-1)!}{(n+1)^2} \).

Step 3: Find \( a_{n+1} \)

To find \( a_{n+1} \), replace \( n \) with \( n+1 \) in \( a_n \): \[ a_{n+1} = \frac{n!}{((n+1)+1)^2} = \frac{n!}{(n+2)^2}. \]

Step 4: Set Up the Ratio \( \frac{a_{n+1}}{a_n} \)

The ratio is: \[ \frac{a_{n+1}}{a_n} = \frac{n!}{(n+2)^2} \times \frac{(n+1)^2}{(n-1)!}. \]

Step 5: Simplify the Ratio

Simplify the expression: \[ \frac{n!}{(n-1)!} = n(n-1), \] so \[ \frac{a_{n+1}}{a_n} = \frac{n(n-1)(n+1)^2}{(n+2)^2}. \]

Step 6: Compute the Limit of the Ratio as \( n \to \infty \)

Calculate the limit: \[ \lim_{n \to \infty} \frac{n(n-1)(n+1)^2}{(n+2)^2}. \]This can be simplified further because the highest degree terms are \( n^3 \) in both the numerator and the denominator: \[ \lim_{n \to \infty} \frac{n^3}{n^2} = \lim_{n \to \infty} n = \infty. \]

Step 7: Analyze the Limit

Since the limit of the ratio is \( \infty \), which is greater than 1, according to the Ratio Test, the series diverges.

Key Concepts

Series Convergence

Series convergence refers to determining whether the sum of an infinite series approaches a finite value as additional terms are added. Understanding this concept is crucial in calculus, as many functions can be expressed as a series.

When discussing series, there are different types of convergence:

• Convergence: A series is convergent if the sum of its terms approaches a specific value as more terms are added. In our exercise, we are tasked with determining convergence using the Ratio Test.
• Divergence: If a series doesn't settle towards a particular number, it is divergent, meaning its sum continues to grow without bound.
• Conditional and Absolute Convergence: A series is absolutely convergent if the sum of the absolute values of the terms converges. If only the original series (without absolute values) converges, it is conditionally convergent.

Analyzing whether a series converges or diverges is often done through tests like the Ratio Test, which we'll explore later in this article. When using these tests, we can determine if summing all the terms will approach a finite number or not.

Absolute Convergence

Absolute convergence is a stronger form of convergence, meaning if a series converges absolutely, it converges regardless of the order of terms. For a series \( \sum a_n \) with terms \( a_n \), it converges absolutely if \( \sum |a_n| \) also converges. This implies the series' behavior is stable enough that even the magnitude alone stands.

When dealing with the Ratio Test, if we can demonstrate absolute convergence, we then ensure the series will behave nicely under various conditions, such as rearrangements of terms. This is due to rearrangement only affecting conditionally convergent series.

• Ensuring absolute convergence provides a stronger guarantee than just regular convergence.
• Failing to show this means the series might be sensitive to rearrangements and is only conditionally convergent, or may not converge at all.
• In our exercise, despite our hopes of establishing absolute convergence via the Ratio Test, we discovered that the result pointed to divergence.

This underscores that not all series achieve this robust form of convergence, explaining why it's so valued in both theoretical and practical mathematical contexts.

Limit Calculation

Limit calculation is at the core of many mathematical concepts, including the Ratio Test. It allows us to determine the behavior of sequences as they approach infinity. When using the Ratio Test, we focus on calculating:

• The limit of the ratio \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), which tells us about the term-to-term growth rate.
• If this limit is less than 1, the series converges absolutely.
• If greater than 1, the series diverges.

In our exercise, we found:

• The calculation simplifies due to equivalent \( n^3 \) growth in numerator and denominator.
• This resulted in an infinite limit, indicating that the original series diverges.

Understanding limit calculations is vital, especially since they often present in simplifying complex mathematical expressions. They can reveal deeper insights into the behavior of series, such as whether an infinite series remains bounded or grows indefinitely. Recognizing these limits forms the backbone of determining convergence through methods like the Ratio Test.