Lim sup an /bn = lim sup an / lim sup bn?

[jem05]

hello,
2 technicality questions:

1) lim sup a_n /b_n = lim sup a_n / lim sup b_n?
2) if lim sup a_n /b_n is finite does that mean
that the sequence {a_n/b_n} bounded?

thank you.

 

[micromass]

1) no. Take (an)=(0,1,0,1,0,1,...) and (bn)=(1,-1,1,-1,1,-1,...).
2) I believe that this is true...

 

[mathman]

2) is false. Let b_n=0 for a finite number of n's.

 

[micromass]

Uh, if a bn=0, then the sequence $$a_n/b_n$$ isn't even good defined. I didnt think the OP meant that...

 

[jem05]

yes ofcourse, i meant bn strictly positive.
ok, thanks a lot

 

[George Jones]

2) if lim sup a_n /b_n is finite does that mean that the sequence {a_n/b_n} bounded?

It depends what "bounded" means.

For example, take b_n = 1 for every n, take a_n = 1/n if n is odd, and take a_n = -n if n is even.

Then, if I'm thinking about this correctly, lim sup a_n /b_n = 0, but a_n /b_n is not bounded from below.