`a_n=(-2/3)^n` Determine whether the sequence with the given n'th term is monotonic and whether it is bounded.

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`a_n=(-2/3)^n=(-1)^n(2/3)^n={(-(2/3)^n if n=2k-1),((2/3)^n if n=2k):}`

This tells us that odd-numbered terms are negative, while even-numbered terms are positive. Since the terms alternate in sign, the sequence is alternating. In other words, the sequence is not monotonic.

All terms of the sequence are in `[-2/3,4/9].`

This is because `-(2/3)^n< -(2/3)^(n+1),` `forall n in NN`  and `lim_(n to infty)-(2/3)^n=0`

Also, `(2/3)^n>(2/3)^n,` `forall n in NN` and `lim_(n to infty)(2/3)^n=0`

Clearly `-2/3` is the smallest term of the sequence and `4/9` is the greatest.

Therefore, the sequence is bounded.

The image below shows the first 20 terms of the sequence. We can see that the sequence is also convergent even though it is not monotonical                             ` `

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