Limit Inferior and Limit Superior of a Sequence with Alternating Terms

[muso07]

Homework Statement

Find liminf(xn) and limsup(xn) for xn = n(1-(-1)^n)

Homework Equations

The Attempt at a Solution

I'm not really getting liminf and limsup, but stumbled through the method in my textbook and got liminf=0, limsup doesn't exist.

Is that right? I don't think I did it right. If it's wrong, I can type out my working (I'd just prefer not to since it's pretty late).

 

[mrbohn1]

You are correct. I fins the easiest definition of limsup to be:

$$\limsup_{n\rightarrow \infty}=\lim_{n\rightarrow \infty}(\sup_{m\geq n}x_m)$$

That is: as n approaches infinity, look at all the "peaks" of the $$x_m$$ for $$m\geq n$$. You are looking for the "last peak".

In the case of this function, the peaks get larger and larger (they occur when n is odd), so the "last one" is infinity, ie. it doesn't exist.

Similarly for liminf, the "troughs" of the function are all zero (when n is even), so the "last one" will also be zero.

I don't know if this will help much...I remember also being very confused by liminf and limsup when I first came across them!

 

[muso07]

It helped a lot, thanks!