Problem 38 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty} \frac{k !}{k^{3} 3^{k}}\)

Step-by-Step Solution

Answer

The series converges.

Step 1: Identify the Ratio Test

Since the series includes both a factorial term and an exponential term, the Ratio/D'Alembert's Test for series convergence is suitable. The Ratio Test says that the absolute value of the ratio of successive terms of the sequence converging to zero will result in the whole series converging.

Step 2: Apply the Ratio Test

Calculate the limit of the ratio of the (n+1)th term to the nth term in absolute values as n tends to infinity: \( \lim_ {n \to \infty} \left| \frac{(n+1)!}{(n+1)^{3} 3^{n+1}} \cdot \frac{n^{3} 3^{n}}{n !} \)

Step 3: Simplify the expression

Simplify the above expression: \( \lim_ {n \to \infty} \left| \frac{n^{3}}{(n+1)^{2}} \cdot \frac{1}{3} \right| \)

Step 4: Evaluate the limit

The above limit evaluates to \(\frac{1}{3}\). According to the Ratio Test, if the absolute value of this limit is less than 1, it indicates convergence of the series.

Step 5: Conclusion

Since the limit of the ratio, which is \( \frac{1}{3} \), is less than 1, it indicates the series converges.

Key Concepts

Ratio Test

The Ratio Test is a popular tool for determining the convergence of a series. It is particularly useful when dealing with sequences that include terms like factorials and exponentials. The test involves calculating the limit of the absolute ratio of successive terms. If this limit, denoted as \( L \), is less than 1, the series converges. However, if \( L \) is greater than 1, the series diverges. If \( L \) equals 1, the Ratio Test is inconclusive.

In practice:

• If \( L < 1 \), the series converges.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test does not give a conclusion.

When using the Ratio Test, carefully simplify the terms of the sequence before plugging them into the limit. This often involves breaking down factorial and exponential terms.

Factorial and Exponential Terms

Factorial and exponential terms frequently appear in series and play a crucial role in determining convergence. Factorials grow very quickly as the number gets larger, much faster than exponential terms. For example, the factorial \( n! \) is the product of all integers from 1 to \( n \), and it increases in size very rapidly.

On the other hand, exponential terms such as \( b^n \) grow at a slower rate than factorials for large \( n \). The growth characteristics of these terms are essential when simplifying terms in series. Specifically, knowing the growth differences helps in applying convergence tests, such as the Ratio Test, effectively.

Limit Evaluation

Limit evaluation is a core component of applying the Ratio Test. It involves finding the limit of a ratio as the series progresses towards infinity. In our example series \( \sum_{k=1}^{\infty} \frac{k!}{k^3 3^k} \), after applying the test, the expression was simplified to \( \lim_{n \to \infty} \left| \frac{n^3}{(n+1)^3} \cdot \frac{1}{3} \right| \).

To evaluate this limit, we examine the simplified expression:

• Observe how \( n^3/(n+1)^3 \) approaches 1 as \( n \to \infty \).
• Multiply by \( 1/3 \) to get the entire limit expression.

The result, \( 1/3 \), indicates that the terms become negligibly small as \( n \) becomes very large, confirming convergence according to the Ratio Test.

Convergence Tests

Convergence Tests offer a systematic way to determine if series converge. Different tests suit different types of series, but in general, they help check if a series reaches a finite sum. Key tests include:

• Ratio Test: As previously discussed, useful for series with factorials or exponentials.
• Root Test: Determines convergence by taking the \( n \)-th root of absolute terms.
• Integral Test: Relies on comparing series to improper integrals for divergence or convergence.
• Comparison Test: Compares terms of one series to another known series.

Using the right test saves time and effort. For any series challenge, understanding the nature of the terms can guide you to the most effective convergence test. When the Ratio Test indicates convergence, it’s often due to the way the terms shrink as they progress.