Examples 4.2.16(c):
The following statements are not equivalent:
Suppose that the first statement is true, i.e.
- There exists an N such that | an+1 / an | 1 for all n > N
- lim sup | an+1 / an | 1
There exists an N such that | a n+1 / a n | 1 for all n > NNow 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
| a n+1 / a n |So, let
- a n = 2 if n is even
- a n = 1 if n is odd
- a n+1 / a n = 1 / 2 if n is even
- a n+1 / a n = 2 / 1 if n is odd