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Chapter 7: Q. 29 (page 604)

For each of the sequences in Exercises 23鈥�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.

29.3-k-2

Short Answer

Expert verified

The sequence is monotonic and bounded and convergent.

The limit of the sequence is-2.

Step by step solution

01

Step 1. Given information

We have been given the sequence3-k-2.

02

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

ak=3-k-2

ak+1-ak=3-k+1-2-3-k-2=3-k-1-3-k=3-k3-1-3-k=3-k13-1=-13k23=-23k+1<0

Thus, ak+1<ak.

The sequence is strictly decreasing. The given sequence is monotonic.

The sequence is bounded above because ak<0for k>0

-2<3-k-2<0

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

The sequence is convergent.

03

Step 3. Determine the limit of the sequence.

limk鈫�鈭�ak=limk鈫�鈭�3-k-2=limk鈫�鈭�13k-2=0-2=-2

Thus, the limit of the sequenceak=3-k-2is-2.

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