Example of Monotone Sequences
In my last post, i explained  the monotone sequences and their criteria of convergency,now i will be a giving solutions to problems of monotone sequences and how to identify if they are decreasing,increasing or constant.
Examples
Evaluate the following sequences and check if they are monotone converging.
(1) an={1−1n}, for n≥1
(2) bn={n3} for n≥1
(3) cn={(−1)n} for n≥1
(4) dn={1n2} for n≥1

Solution:
(1) To evaluate all of the given sequences above we use the monotone formula which is nth=an+1−an.
Now let an={1−1n} and an+1={1−1n+1}.
nth=[1−1n+1]−[1−1n],take the L.C.M of both sides=n+1−1n+1−[n−1n]=nn+1−[n−1n]Take the L.C.M=n2−(n+1)(n−1)n(n+1)=n2−(n2−1)n(n+1)=1n(n+1) for all n≥1choose n=11n(n+1)=12n=21n(n+1)=16n=31n(n+1)=112hence it is strictly monotone increasing,because1n(n+1)>0
(2) bn={n3} for n≥1
using nth=bn+1−bn, let bn=n3 and bn+1=(n+1)2
=n3+3n2+3n+1−n3=3n2+3n+3 forn≥1n=1=3(1)2+3(1)+3=9n=23n2+3n+3=21n=33n2+3n+3=39hence it is monotone strictly increasing because3n2+3n+3>0
(3) cn={(−1)n} for all n≥1
using nth=cn+1−cn
=(−1)n+1−(−1)n=(−1)n.(−1)−(−1)nfor n=1(−1)n.(−1)−(−1)n=2n=2(−1)n.(−1)−(−1)n=−2n=3(−1)n.(−1)−(−1)n=2

from here we can see that the nth values of (−1)n.(−1)−(−1)n are not unique in the sense that some are positive and others are negative.hence cn={(−1)n} is not a monotone sequence.
(4) dn={1n2} for n≥1
using nth=dn+1−dn
1(n+1)2−1n2=n2−(n+1)2n2(n+1)2=n2−(n2+2n+1)n2(n+1)2=−2n−1n2(n+1)2n=1−2n−1n2(n+1)2=−34n=2−2n−1n2(n+1)2=−536n=3−2n−1n2(n+1)2=−7144

Hence we can conclude that this monotone is strictly decreasing because −2n−1n2(n+1)2<0 for all n≥1.