- #1

- 11

- 0

## Homework Statement

find the limsup and liminf of [tex]a_{n}=(-1)^{n}\left( 1+\frac{1}{n} \right)[/tex]

## Homework Equations

Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup is the limit of

[tex]a_{n}=sup_{(k>=n)}a_{k}[/tex]

## The Attempt at a Solution

I started by using the identity sup(f+g) = sup(f) + sup(g) to turn the initial equation into a sum after multiplying it out. Then I separated the problem into two different limits. Now it seems obvious that the limsup of the given equation is 1 but I'm not sure how to prove that result.

limsup[tex](-1)^{n}[/tex] is clearly one. I showed this by setting n=2k, for some natural number k, and then saying for some n > k, [tex](-1)^{n}=1[/tex]. Then the sup is 1, and the limsup is 1 also.

I'm running into problems finding the limsup[tex]\frac{(-1)^{n}}{n}[/tex].

Do I have to use the same n = 2k, because the limit here goes to 0, leaving me with the result I want, but I'm not sure that: a) how to go about writing this out and b) if I'm even proceeding correctly.

I think if I can work through the limsup portion of the problem, it will be easy to apply the reasoning to the liminf portion.

Thanks.