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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.

1+1kk

Short Answer

Expert verified

The given sequence is monotonic, bounded and convergent.

The limit of the sequence is e.

Step by step solution

01

Step 1. Given Information  

We are given the sequence 1+1kkand 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=1+1kk.

The ratio

ak+1ak=1+1k+1k+11+1kk=k+2k+1k+1k+1k+1k+1kk+1=k+2k+1.kk+1k.k+2k+1>1(k>0)ak+1>ak

The sequence is strictly increasing so it is monotonic.

03

Step 3. Finding bounded  

The sequence 1+1kkis bounded below because 2<ak. As k>1,ak3. The decreasing sequence has a lower bound and is 2and an upper bound 3.

Therefore, the given sequence is bounded.

04

Step 4. Finding the limit 

The monotonic decreasing sequence is bounded below and hence convergent.

limkak=limk1+1kk=e

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