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Chapter 7: Q. 66 (page 593)

Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

ak=k2k!

Short Answer

Expert verified

The given sequence is monotonic and unbounded.

Step by step solution

01

Step 1. Given Information.

The given sequence isak=k2k!.

02

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will use the ratio test.

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

So, the ak+1term is

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

According to the ratio test,

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

Now, ak+1ak<1fork鈮�2.

Thus, the sequence is eventually decreasing for k鈮�2,and it is a monotonic sequence.

03

Step 3. Determine whether the given sequence is bounded or unbounded.   

As we have shown that the sequence is eventually decreasing for k鈮�2.Thus, it has an upper bound of 2but it doesn't have a lower bound, so it is an unbounded sequence.

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