## 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*| a*for all_{n+1}/ a_{n}| 1*n > N* *lim sup | a*_{n+1}/ a_{n}| 1

There exists anNow recall the definition of theNsuch that| afor all_{n+1}/ a_{n}| 1n > N

*lim sup*as the limit of the supremums of the truncated sequences:

But iflim sup | a_{n+1}/ a_{n}| = lim ( sup{ | a_{n+1}/ a_{n}| , j n } )

*n > N*, then the expression

*| a*. Therefore, the

_{n+1}/ a_{n}| 1*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

Here the2, 1/2, 2, 1/2, 2, 1/2, ...

*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+1}/ a_{n}|

*a*if_{n}= 2*n*is even*a*if_{n}= 1*n*is odd

*a*if_{n+1}/ a_{n}= 1 / 2*n*is even*a*if_{n+1}/ a_{n}= 2 / 1*n*is odd