How study margins of sequence a subscript n=(2n+1)/(n+5)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to study the boundaries of the sequence `a_n = (2n+1)/(n+5)` , hence, you may start by finding the upper boundary, such that:

`a_n = (2n+1)/(n+5)`

Using reminder theorem yields:

`a_n = 2 - 9/(n + 5)`

You should notice that the members of the sequence are above bounded by 2, hence, the upper boundary is sup `{a_n} = 2` .

You may also notice that all the terms of the sequence are positive, hence, the sequence is inferior bounded by inf `{a_n} = 0` .

Hence, evaluating the boundaries of the given sequence, considering the completeness axiom, yields inf `'{a_n} = 0` and sup `{a_n} = 2` .


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