Determine if a sequence {an} is monotonic, bounded, convergent

[sara_87]

Determine whether the sequence {an} defined below is
(a) monotonic
(b) bounded
(c) convergent and if so determine the limit.

(1) {an}=(sqrt(n))/1000
a) it is monotonic as the sequence increase as n increases.
b) it's not bounded (but I'm not sure why)
c) divergent since limit doesn't exit (tends to infinity)
is this correct?
(2) {an}=(-2n^2)/((4n^2)-1)
i don't know how to answer parts (a) (b) and (c) to this one can someone help me please.
thank you

 

[Gib Z]

1) a) Correct.

b) A sequence is bounded if |a_n| is less than some real number M, for all values of n. Does this help? Is there such a number M in this case?

c) Correct. Also, an unbounded monotonic increasing sequence necessarily diverges to positive infinity. It may be a good exercise to prove why.

2) a) Trick is to subtract a half and then add it again, like this;

$$a_n = - \left( \frac{2n^2}{4n^2 -1} \right) = - \left( \frac{2n^2 - \frac{1}{2} }{4n^2 -1} + \frac{1}{2(4n^2-1)}\right) = - \left ( \frac{1}{2} + \frac{1}{2(4n^2-1)}\right)$$.

Now use partial fractions, it should be much easier then =]

PS - I noticed its been about one and a half days since you've posted this, and yet received no help. This is probably because you placed this in the wrong forum - Put it in the Homework Help forums (Calculus and Beyond) next time and you should get help much faster.

 

[dodo]

A common trick with powers in a fraction is to divide up and down by the highest power, like this:

$$a_n = -\frac{\frac{2 n^2}{n^2}}{\frac{4 n^2 - 1}{n^2}} = -\frac 2 {4 - \frac 1 {n^2}}$$​

Now it's easy to see what the limit is.

 

[Gib Z]

Shoot me that is a much easier way :(