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Chapter 10: Problem 20

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{n !}{3^{n}} $$

Short Answer

Expert verified
The series diverges.

Step by step solution

01

Express the General Term

The general term of the series here is \( a_{n} = \frac{n !}{3^{n}} \)
02

Find the Ratio Between Terms

The ratio between the (n+1)th term and nth term can be calculated as \( R = \frac{a_{n+1}}{a_{n}} = \frac{\frac{(n+1) !}{3^{n+1}}}{\frac{n !}{3^{n}}} = \frac{(n+1)}{3} \)
03

Apply Limit and Use Ratio Test Criteria

The Ratio Test states that if the limit as n approaches infinity of the absolute value of the ratio of consecutive terms is less than 1, the series converges. Let's apply this criteria by finding the limit, \( L = \lim_{n \rightarrow \infty} |R| = \lim_{n \rightarrow \infty} \frac{n+1}{3} = \infty \). Because this limit is greater than 1, according to the Ratio Test, the series diverges.

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Most popular questions from this chapter

Sales A company produces a new product for which it estimates the annual sales to be 8000 units. Suppose that in any given year \(10 \%\) of the units (regardless of age) will become inoperative. (a) How many units will be in use after \(n\) years? (b) Find the market stabilization level of the product.

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Determine whether the sequence is arithmetic or geometric, and write the \(n\) th term of the sequence. $$ 20,10,5, \frac{5}{2}, \dots $$
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