Example:
The following statements are not equivalent:
- There exists an N such that |
/
| 
1 for all n > N
- lim sup |/ | 1
In fact, the first statement implies the second, but not the other
way around.
Suppose that the first statement is true, i.e.
- There exists an N such that
| a n+1 / a n |
1 for all n > N
Now recall the definition of the lim sup as the limit of the supremums
of the truncated sequences:
- lim sup | a n+1 / a n | =
lim ( sup{ | a n+1 / a n | ,
j n } )
But if n > N, then the expression
| a n+1 / a n |
1.
Therefore, the lim sup must also be greater than one.
As an example to show that the second statement does not imply the first
one, consider the sequence
- 2, 1/2, 2, 1/2, 2, 1/2, ...
Here the lim sup is clearly equal to 2, but there is no N
such that the terms are all greater than or equal to 1 for n > N.
What remains for us to do is write this sequence as a quotient
So, let
- a n = 2 if n is even
- a n = 1 if n is odd
Then
- a n+1 / a n = 1 / 2 if n is even
- a n+1 / a n = 2 / 1 if n is odd
Therefore, the second statement above does not imply the first one.
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