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Chapter 7: Q. 54 (page 592)

In Exercises 51鈥�54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{1}{k!}\}$

Short Answer

Expert verified

The given sequence is strictly decreasing for $k 鈮� 1 .$

Step by step solution

01

Step 1. Given Information.

The given sequence is $\{\frac{1}{k!}\} .$

02

Step 2. Use the difference test.

To analyze the monotonicity of the given sequence we will use the difference test.

Let the general term of the sequence is $a_{k} = \frac{1}{k!} .$

So, the term $a_{k + 1}$ is

$a_{k + 1} = \frac{1}{(k + 1)!} .$

According to the test,

$a_{k + 1} - a_{k} = \frac{1}{(k + 1)!} - \frac{1}{k!} = \frac{k! - (k + 1)!}{(k)! (k + 1)!} = \frac{k! - (k + 1) k!}{(k)! (k + 1)!} = \frac{k! (1 - k - 1)}{(k)! (k + 1)!} = \frac{- k}{(k + 1)!}$

Now, $a_{k + 1} - a_{k} 鈮� 0 for all k 鈮� 1 .$

Thus, the sequence is strictly decreasing for $k 鈮� 1 .$

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Most popular questions from this chapter

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

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Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

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