Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\sum_{n=1}^{\infty} \frac{n !}{5+n}\)

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty} \frac{n!}{5+n}\) diverges.

Step 1: Identify the Series

We need to determine the convergence or divergence status of the series \(\sum_{n=1}^{\infty} \frac{n!}{5+n}\). This is an infinite series starting from \(n=1\) and does not have an upper bound for \(n\).

Step 2: Apply the Ratio Test

The Ratio Test is useful for series involving factorials. The Ratio Test evaluates \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|\). So, we consider:\[ a_n = \frac{n!}{5+n}, \]\[ a_{n+1} = \frac{(n+1)!}{5+(n+1)} = \frac{(n+1) \cdot n!}{6+n} \]Calculate:\[\frac{a_{n+1}}{a_n} = \frac{(n+1) \cdot n!}{(6+n)} \cdot \frac{5+n}{n!} = \frac{n+1}{6+n}\]

Step 3: Evaluate the Limit of the Ratio

Now compute:\[\lim_{{n \to \infty}} \frac{n+1}{6+n} = \lim_{{n \to \infty}} \frac{n+1}{n \cdot (1 + \frac{6}{n})} = \lim_{{n \to \infty}} \frac{1 + \frac{1}{n}}{1 + \frac{6}{n}}\]As \(n \to \infty\), we have both \(\frac{1}{n} \to 0\) and \(\frac{6}{n} \to 0\), resulting in:\[\lim_{{n \to \infty}} \frac{1}{1} = 1\]

Step 4: Conclude with the Ratio Test Result

According to the Ratio Test, if the limit equals 1, the test is inconclusive. Thus, we have learned the Ratio Test does not prove the convergence or divergence of the given series.

Step 5: Check Growth of Factorials versus Linear Terms

Even though the Ratio Test is inconclusive, observe that \(n!\) grows exponentially while \(5+n\) grows linearly. Typically, \(n!\) outpaces linear factors, causing the series \(\sum \frac{n!}{5+n}\) to diverge.

Key Concepts

Convergence and Divergence of Series

In mathematics, understanding whether an infinite series converges or diverges is crucial. A series is a sum of terms of a sequence. When investigating convergence, we want to know if these terms approach a distinct number as we add infinitely many of them. On the contrary, a series diverges if the sum keeps growing indefinitely without approaching a fixed total.

For instance, when working with the series \( \sum_{n=1}^{\infty} \frac{n!}{5+n} \), we examine if the combined sum of all terms approaches a specific value.

• If a series converges, adding more terms progressively brings the total closer to a certain number.
• If it diverges, adding more terms results in it increasing unboundedly or oscillating indefinitely.

Hence, determining convergence or divergence helps us understand the fundamental behavior of the series.

Factorial Growth

Factorials, represented as \(n!\), express the product of all positive integers up to \(n\). Their growth rate is significant in calculus and series analysis.

Factorials grow astonishingly fast, much more rapidly than polynomial or even exponential functions.

• For example, \(5! = 1 \times 2 \times 3 \times 4 \times 5 = 120\).
• As \(n\) increases, \(n!\) quickly becomes a very large number.

This swift growth rate can affect a series' convergence. In our series \( \sum_{n=1}^{\infty} \frac{n!}{5+n} \), the factorial in the numerator grows much faster than the linear term \(5+n\) in the denominator. Such disproportionate growth often leads to divergence because the terms of the series become large, not diminishing to approach a zero sum.

Infinite Series Analysis

Analyzing infinite series involves evaluating the long-term behavior of its terms and cumulative sums. Various methods and tests are employed to determine the series' nature.

• The Ratio Test is one kind of check, especially for series including factorial elements.
• Other tests include the Comparison Test, Integral Test, and Alternating Series Test.

For our specific series, the Ratio Test was applied by examining the behavior as \(n\) approaches infinity. Although the Ratio Test returned an inconclusive result, it set the stage for further thinking about factorial growth compared to linear functions, directing us to the likely divergence of the series.

Limit Evaluation in Calculus

Limits are the mathematical way of exploring values that a function or sequence approaches as the input gets arbitrarily large or small. Evaluating limits is a key technique in calculus to deduce behaviors of sequences and functions.

In the given series, the Ratio Test required calculating\[\lim_{{n \to \infty}} \frac{n+1}{6+n}\]This involved simplifying the expression to assess its behavior as \(n\) becomes very large.

• By breaking it down, the expression tends toward \(\frac{1}{1} = 1\).
• This value indicates that the Ratio Test is inconclusive and requires alternative approaches or tests.

The limit evaluation highlights that while the value helps test the convergence, it should be combined with further analysis or different tests to draw a definitive conclusion about an infinite series.