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For each of the sequences in Exercises 23–52 determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.

k!

Short Answer

Expert verified

The given sequence is monotonic, bounded below and not convergent.

Step by step solution

01

Step 1. Given Information 

We are given the sequence k!and we need to find if the sequence is monotonic, bounded and the limit if it is convergent.

02

Step 2. Finding monotonic 

The general term is ak=k!.

The ratio

ak+1ak=k+1!k!=k+1k!k!=k+1>1ak+1>ak

The sequence is strictly increasing so it is monotonic.

03

Step 3. Finding bounded  

The sequence is bounded below because 1ak. The increasing sequence has a lower bound and is 1and no upper bound.

Therefore, the given sequence is bounded below.

04

Step 4. Finding the limit  

The monotonic increasing sequence has no upper bound.

Hence, it is not convergent.

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