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Chapter 7: Q. 24 (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.

24.1+1k

Short Answer

Expert verified

The sequence is monotonic and bounded and the convergent.

The limit of the sequenceak=1+1kis1.

Step by step solution

01

Step 1. Given information

We have been given the sequence1+1k.

02

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

1k>1k+11+1k>1+1k+1ak>ak+1

The sequence is decreasing sequence and hence is monotonic.

The lower bound of the sequence ak=1+1kis 0.

Thus the sequence is bounded below.

The monotonic decreasing sequence with lower bound is convergent.

03

Step 3. Determine the limit of the sequence.

limk鈫�鈭�ak=limk鈫�鈭�1+1k=1+0=1

Thus, the limit of the sequenceak=1+1kis1.

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Most popular questions from this chapter

Consider the series

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