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Chapter 7: Q. 55 (page 592)

In Exercises 55鈥� 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

k2k!

Short Answer

Expert verified

The given sequence is eventually decreasing fork鈮�2.

Step by step solution

01

Step 1. Given Information.

The given sequence isk2k!.

02

Step 2. Use the ratio test.

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

Let the general term of the sequence is ak=k2k!.

So, the term ak+1is

ak+1=k+12k+1!.

According to the ratio test,

ak+1ak=k+12k+1!k2k!=k!k+12k2k+1!=k!k+12k2k+1k!=k+1k2

Now, ak+1ak<1fork鈮�2.

Thus, the given sequence is eventually decreasing fork鈮�2.

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