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

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`a_n=n e^(-n/2)`

Monotonicity

First  five terms of the sequence are

`a_1=e^(-1/2)=0.6065`

`a_2=2e^-1=0.7358`

`a_3=3e^(-3/2)=0.6693`

`a_4=4e^-2=0.5413`

`a_5=5e^(-5/2)=0.4104`

We can see that after the second term, the terms are decreasing so it is possible that the whole sequence is monotonically decreasing. Let us verify that.

`a_n>a_(n+1)`

`n e^(-n/2)>(n+1)e^(-(n+1)/2)=(n+1)e^(-n/2)e^(-1/2)`

Divide the inequality by `e^(-n/2).` We can do that because `e^(-n/2)>0,` `forall n in NN.`   ` `

`n>(n+1)e^(-1/2)=n e^(-1/2)+e^(-1/2)`

Divide by `n.` We can do that because `n>0.`

`1>e^(-1/2)+e^(-1/2)/n`

Since `e^(-1/2)<1,` we can find sufficiently large `n` such that the above inequality holds. In this case `n=2.`

`1>e^(-1/2)+e^(-1/2)/2=0.9098`

Therefore, `forall n geq 2` `(a_n>a_(n+1))` which means that the sequence is monotonically decreasing.

Boundedness

We have shown that the sequence is monotonically decreasing from second term onwards. This means that the second terms is also maximum of the sequence. In other words the sequence is bounded from above by `a_2=2e^-1.`

On the other hand if we look at the sequence `a_n=n e^(-n/2),` we see that all of its terms are positive. This is because `n>0` and exponential function is always positive so `e^(-n/2)>0` and the product of two positive numbers is itself positive.

Therefore, `a_n>0,` `forall n in NN` i.e. the sequence is bounded by zero from below.

We can conclude that `forall n in NN,` `a_n in [2e^(-1),0)` i.e. the sequence is bounded from both below and above.                                                         

The image below shows first 20 terms of the sequence. Both boundedness and monotonicity can clearly be seen on the image.                    

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