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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.

k2110000k+2

Short Answer

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Ans:k2110000k+2

Step by step solution

01

Step 1. Given information.

given,

k2110000k+2

02

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.   

In the sequence {ak}=k2110000k+2the gentral term is ak=k2110000k+2.

03

Step 3.  The general term of the sequence is ak=k2−110000k+2.

The term ak+1-akgives

ak+1ak=(k+1)2110000(k+1)+2k2110000k+2(Substitution)=k2+2k+1110000k+10000+2k2110000k+2=k2+2k10000k+10002k2110000k+2(Simplify)=k2+2k(10000k+2)k21(10000k+10002)(10000k+10002)(10000k+2)=10000k2+10004k+10002(10000k+10002)(10000k+2)>0(Fork>0)Thus,ak+1>ak

The sequence {ak}=k2110000k+2strictly increases. Therefore, the given sequence is monotonic.

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