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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.

27.-1kk+6

Short Answer

Expert verified

The sequence is not monotonic and bounded and convergent.

The limit of he sequence is0.

Step by step solution

01

Step 1. Given information

We have been given the sequence-1kk+6.

02

Step 2. Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below.

The sequence ak=-1kk+6is not a monotonic sequence because sign of -1kk+6varies as kincreases.

The sequence ak=-1kk+6is a bounded sequence because

0k17for k>0

The given sequence has upper and lower bounds, therefore, the sequence is bounded.

03

Step 3. Determine the limit of the sequence.

limkak=limk-1kk+6=0

The sequence is not monotonic but is bounded. Therefore, the sequence converges to0.

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