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Chapter 7: Q. 53 (page 604)

Determine whether the sequence converges or diverges. If the sequence converges, give the limit.

(k!)1/k

Short Answer

Expert verified

Ans: The sequence{ak}=(k!)1/kis divergent.

Step by step solution

01

Step 1. Given information.

given,

(k!)1/k

02

Step 2. The objective is to determine whether the sequence is convergent or divergent and to find the limit of the sequence if the sequence is convergent.

In the sequence {ak}=(k!)1/kthe general term is ak=(k!)1/k

The ratio ak+1akgives

role="math" localid="1649302149750" ak+1ak=((k+1)!)1/k+1(k!)1/k>1(Fork>0)Thus,ak+1>ak

The sequence {ak}=(k!)1/kis strictly increasing. The given sequence is monotonic.

03

Step 3. The sequence {ak}=(k!)1/k is bounded below because

0<akfor k>0

The sequence is an increasing sequence and doesn't have any upper bound.

The given sequence has a lower bound, therefore, the sequence is bounded below.

04

Step 4. The monotonic increasing sequence is bounded above is convergent. 

The monotonic decreasing sequence {ak}=(k!)1/kis not bounded above and hence is not convergent. Therefore, the given sequence is divergent.

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