You may consider a sequence
Now you may examine
how it behaves around zero.
Fix , take and define
I know this sequence should alternate about . I.E., the odd terms are greater than and the even terms are less than . However, I cannot seem to show this.
I've tried to use the fact that and but I can't seem to get anything to work.
I would appreciate a hint very much.
The recursive relation can be written as...
Supposing there is an 'attractive fixed point' at and the condition for monotonic convergence at is that...
It is easy to verify that the condition (2) is satisfied for so that in this case the sequence will be increasing if and decreasing if . For the sequence remains convergent but is 'oscillating'...