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

  • Thread starter jem05
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  • #1
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hello,
2 technicality questions:

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

thank you.
 

Answers and Replies

  • #2
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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...
 
  • #3
mathman
Science Advisor
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2) is false. Let bn=0 for a finite number of n's.
 
  • #4
22,089
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Uh, if a bn=0, then the sequence [tex]a_n/b_n[/tex] isn't even good defined. I didnt think the OP meant that...
 
  • #5
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yes ofcourse, i meant bn strictly positive.
ok, thanks a lot
 
  • #6
George Jones
Staff Emeritus
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2) if lim sup an /bn is finite does that mean that the sequence {an/bn} bounded?
It depends what "bounded" means.

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

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

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