Problem 3 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The series diverges by the Ratio Test.

Step 1: Identify the series terms

The series provided is \( \sum_{n=1}^{\infty} a_n \), where \( a_n = \frac{(n-1)!}{(n+1)^2} \).

Step 2: Apply the Ratio Test

The Ratio Test involves evaluating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). This will determine whether the series converges absolutely or diverges.

Step 3: Find the ratio \( \frac{a_{n+1}}{a_n} \)

Calculate \( a_{n+1} = \frac{n!}{(n+2)^2} \). Then, find the ratio:\[\frac{a_{n+1}}{a_n} = \frac{\frac{n!}{(n+2)^2}}{\frac{(n-1)!}{(n+1)^2}} = \frac{n! \cdot (n+1)^2}{(n-1)! \cdot (n+2)^2}\]

Step 4: Simplify the ratio expression

Simplify \( \frac{n! \cdot (n+1)^2}{(n-1)! \cdot (n+2)^2} \):- The \( n! \) and \( (n-1)! \) cancel to \( n \).- Rewrite as \( \frac{n \cdot (n+1)^2}{(n+2)^2} \).So, \( \frac{a_{n+1}}{a_n} = \frac{n \cdot (n+1)^2}{(n+2)^2} \).

Step 5: Evaluate the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

Find the limit:\[\lim_{n \to \infty} \frac{n(n+1)^2}{(n+2)^2} = \lim_{n \to \infty} \frac{n(n^2 + 2n + 1)}{n^2 + 4n + 4} = \lim_{n \to \infty} \frac{n^3 + 2n^2 + n}{n^2 + 4n + 4}\]Divide numerator and denominator by \( n^2 \):\[\lim_{n \to \infty} \frac{n + \frac{2}{n} + \frac{1}{n^2}}{1 + \frac{4}{n} + \frac{4}{n^2}} = \lim_{n \to \infty} n = \infty\]Thus, the limit is \( \infty \).

Step 6: Conclusion of the Ratio Test

The result of the Ratio Test shows that since the limit is \( \infty \) (greater than 1), the series diverges.

Key Concepts

Series Convergence

Understanding series convergence is crucial when dealing with infinite series. A series converges if the sum of its infinite terms results in a finite number. Conversely, if the sum grows indefinitely, the series is considered to diverge.

There are various tests available to examine convergence, and each test applies to specific types of series. Some popular tests include:

• Ratio Test: This test is particularly useful for series with factorials and exponentials. It's applied by evaluating the limit of the ratio of consecutive terms.
• Root Test: Similar to the Ratio Test but involves taking the nth root of terms.

Using these tests, you can efficiently determine whether a series converges or diverges.

In the given exercise, the Ratio Test was used, revealing that the series diverges due to the limit evaluation surpassing 1.

Factorial Series

Factorial series often involve terms with factorials like \( n! \) (n factorial), which are pivotal in advanced calculus and combinatorics. Factorials grow very rapidly, which impacts convergence of the series they are part of. In factorial expressions, each integer up to the given number is multiplied together.
For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Because of this rapid growth, factorials generally lead to divergent series unless the terms involve compensating factors in the denominator.

• A common approach to handle these series is to use tests that simplify expressions, like the Ratio Test.
• The n factorial usually results in dividing terms to simplify complicated series.

In the exercise, \( (n-1)! \) encountered in the series expression was crucial. It's simplified using properties of factorials during the Ratio Test.

Limit Evaluation

Limit evaluation is the process of finding the behavior of a function or sequence as the input approaches some value, often infinity. When evaluating series for convergence, determining the limit of ratios of consecutive terms is key.
For a series, if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), the series converges absolutely. If this limit is greater than 1, the series diverges.

• During limit evaluation, indeterminate forms often arise, requiring techniques like L'Hôpital's Rule or polynomial division for simplification.
• It's also essential to understand the hierarchy of terms as they grow; higher powers of n tend to dominate lower ones.

In the exercise, the evaluation resulted in an infinite limit, highlighting the divergence of the series since the growth rate exceeded 1.