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الكلية كلية التربية للعلوم الصرفة     القسم قسم الفيزياء     المرحلة 2
أستاذ المادة هدى عامر هادي       21/03/2021 20:36:31
Increasing, Decreasing, and Bounded Sequences
In the previous lesson we studied the sequences and we understood the sequences limits, the convergence, and the divergence. In this lesson we will study the increasing, decreasing, and bounded sequences.
Def 1 : A sequence {a_n } is called an increasing sequence (non-decreasing) if a_n?a_(n+1) ?n. In another word a_1?a_2?a_3??.
Def 2 : A sequence {a_n } is called decreasing sequence (non-increasing) if a_n?a_(n+1) ?n. In another word a_1?a_2?a_3??.


Ex: Test the increasing and decreasing of the following sequences:
? 1.{(n^2+2n+1)/(?3n?^2+n)}?_(n=1)^? 2. {?tan?^(-1) (n)}_(n=1)^?
Sol. 1. a_n=(n^2+2n+1)/(?3n?^2+n)=(n(n+2)+1)/(n(3n+1))=(n+2)/(3n+1)+1/(n(3n+1)).
Let’s prove (n+2)/(3n+1) is a decreasing or increasing sequence.
decreasing:-
a_n>a_(n+1)?(n+2)/(3n+1)>(n+1+2)/(3(n+1)+1)>(n+3)/(3n+4)

?(n+2)(3n+4)>(n+3)(3n+1)
?3n^2+10n+8>3n^2+10n+3
?8>3.
? (n+2)/(3n+1) is decreasing sequence and we know that 1/(n(3n+1)) is also decreasing.
So ? {(n^2+2n+1)/(?3n?^2+n)}?_(n=1)^? is decreasing sequence.

Sol. 2. a_n=?f(n)=tan?^(-1) (n)
? f?^ (n)=1/(n^2+1)>0 for ?n?1.
? {?tan?^(-1) (n)}_(n=1)^? is increasing sequence.

Def 3 : If {a_n } an increasing or decreasing sequence then is called monotonic sequence.
Def 4 : A sequence {a_n } is bounded from above if ? M?R s.t. a_n?M ?n, the number M is an upper bounded for {a_n }. If M is upper bounded for {a_n }, but no number less than M is an upper bounded for {a_n }, then M is the least upper bounded for {a_n }.
Def 5 : A sequence {a_n } is bounded from below if ? m?R s.t. a_n?m ?n, the number m is lower bounded for {a_n }. If m is lower bounded for {a_n }, but no number greater than m is a lower bounded for {a_n }, then m is the greatest lower bounded for {a_n }.
Def 6 : If {a_n } is bounded from above and below, then {a_n } is bounded. If {a_n } is not bounded, then we say that {a_n } is an unbounded sequence.
Note: The sequence to be increasing it must be increasing for all n. This means the all terms have to be increasing. Also, the sequence is monotonic, if it is increasing only or decreasing only.
Theorem 1: If a sequence {a_n } is both bounded and monotonic, then the sequence converges.
Theorem 2: If a sequence {a_n } is bounded from above and increasing, then the sequence converges.
Theorem 3: If a sequence {a_n } is bounded from below and decreasing, then the sequence converges.


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