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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.

25.sinkπ2

Short Answer

Expert verified

The sequence is not monotonic and bounded and not convergent.

The sequence has no limit.

Step by step solution

01

Step 1. Given information

We have been given the sequencesinkπ2.

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=sinkπ2is not a monotonic sequence because sign of sinkπ2varies as kincreases.

The sequence ak=sinkπ2is a bounded sequence because

-1sinkπ21

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

The sequence is not monotonic but is bounded. Therefore, the sequence is not convergent and has no limit.

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