Problem 33 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{(n+1)(n+2)}{n !} $$

Step-by-Step Solution

Answer

The series converges by the Limit Comparison Test.

Step 1: Identify the series

The given series is \( \sum_{n=1}^{\infty} \frac{(n+1)(n+2)}{n!} \). This series involves factorials in the denominator.

Step 2: Consider the Ratio Test

The Ratio Test is useful for series with factorials. It involves calculating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{(n+1)(n+2)}{n!} \).

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

The term \( a_{n+1} \) is \( \frac{((n+1)+1)((n+1)+2)}{(n+1)!} = \frac{(n+2)(n+3)}{(n+1)!} \).

Step 4: Compute \( \frac{a_{n+1}}{a_n} \)

This ratio is \( \frac{\frac{(n+2)(n+3)}{(n+1)!}}{\frac{(n+1)(n+2)}{n!}} = \frac{(n+2)(n+3) \cdot n!}{(n+1)! \cdot (n+1)(n+2)} = \frac{(n+3)n!}{(n+1)! \cdot (n+1)} = \frac{n+3}{n+1} \).

Step 5: Calculate the Limit of the Ratio

Find \( \lim_{n \to \infty} \frac{n+3}{n+1} = \lim_{n \to \infty} \frac{1 \cdot n + 3}{1 \cdot n + 1} = 1 \).

Step 6: Conclude with the Ratio Test Result

The Ratio Test is inconclusive when the limit is 1, so this test doesn't determine convergence or divergence.

Step 7: Consider the Direct Comparison or limit comparison Test

Compare \( \frac{(n+1)(n+2)}{n!} \) to the series \( \sum \frac{n^2}{n!} \). The series \( \frac{n^2}{n!} \) converges by comparison to the exponential series.

Step 8: Apply the Limit Comparison Test

Calculate \( \lim_{n \to \infty} \frac{\frac{(n+1)(n+2)}{n!}}{\frac{n^2}{n!}} = \lim_{n \to \infty} \frac{(n+1)(n+2)}{n^2} = \lim_{n \to \infty} \frac{n^2 + 3n + 2}{n^2} = 1 \).

Step 9: Determine the Series' Behavior

Since \( \frac{n^2}{n!} \) converges and \( \frac{(n+1)(n+2)}{n!} \) is similar, the original series \( \sum \frac{(n+1)(n+2)}{n!} \) converges by the Limit Comparison Test.

Key Concepts

Ratio Test

In calculus, the Ratio Test is a handy tool to determine whether a series converges or diverges, especially when factorials or exponential expressions are involved. To use the Ratio Test, we examine the limit of the absolute value of the ratio of sequential terms in the series. The test is applied by calculating \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

There are three possible outcomes:

• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In the exercise we considered, the Ratio Test was inconclusive because the limit equaled 1. This means that the series could neither be confirmed as convergent nor divergent just by this test.

Factorial Series

Factorial series involve terms that include factorials, typically in the denominator. Factorials grow extremely quickly, which often suggests that such series may converge due to the swift increase in the size of the denominator.

In the exercise, the series in question was \( \sum_{n=1}^{\infty} \frac{(n+1)(n+2)}{n!} \). Here, the factorial \( n! \) in the denominator plays a crucial role in influencing the convergence of the series. Often, when using the Ratio Test or Limit Comparison Test, the presence of a factorial can significantly simplify the analysis by accelerating the decline of term sizes, which is favorable for convergence.

Limit Comparison Test

The Limit Comparison Test is another technique used to test the convergence of a series. This test is particularly useful for series where the Ratio Test is inconclusive. It involves comparing the series with another one whose convergence behavior is known.

To apply the test, compute the limit:\[\lim_{n \to \infty} \frac{a_n}{b_n}\]where \( a_n \) represents the terms of the series of interest, and \( b_n \) represents the terms of a known convergent or divergent series. If this limit is positive and finite, then both series will either converge or diverge together.

In this exercise, comparing the series \( \sum \frac{(n+1)(n+2)}{n!} \) to \( \sum \frac{n^2}{n!} \) helped establish convergence since \( \sum \frac{n^2}{n!} \) is analogous to an exponential series, which converges.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Understanding the behavior of infinite series is central in calculus as it reveals important properties of functions and their approximations.

The convergence or divergence of an infinite series depends on whether the series approaches a finite limit as more terms are added. Various tests, such as the Ratio Test and Limit Comparison Test, are used to examine this behavior.

The formula for an infinite series is written as:\[\sum_{n=1}^{\infty} a_n\]where each \( a_n \) is a term of the series. Verifying convergence can assist in identifying the sum or the solution to real-world problems.

• If a series converges, it has a finite sum.
• If it diverges, the sum is infinite or undefined.

In our exercise, the series converged, illuminating a fundamental trait of many factorial series where the rapidly growing factorial term leads to convergence.