I think You couldn't. Supposing that is...
(1)
... the behaviour of can be 'oscillating' around 1 and in this case it isn't 'for where N is some fixed integer'...
Marry Christmas from Serbia
This is from so called ratio test, which says
The series Ʃa(n) diverges if abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer.
I was wondering if I could replace the condition with lim inf abs{a(n+1)/a(n)} ≥ 1.
So basically what I'm asking is whether these two below are equivalent or not.
1. abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer
2. lim inf abs{a(n+1)/a(n)} ≥ 1
*a(n) means nth term of the sequence, and abs means absolute value.
Thank you in advance.
Thank you for the reply first.
The example you took shows even if 2 is satisfied, 1 need not be satisfied, so clearly they are not equivalent.
What about the other way around?
Can I conclude that if 1 is satisfied, 2 is satisfied?
or is there an example which satisfies 1 but doesn't satisfy 2?
Can I conclude that if 1 is satisfied, 2 is satisfied?... or is there an example which satisfies 1 but doesn't satisfy 2? ...
Let's indicate and consider the possibility . Clearly 1) is satisfied and 2) is not because the limit for n tending to infinity of doesn't exist...
Marry Christmas from Serbia
Oh may be you misunderstood what I wrote as 'lim inf'. I didn't mean limit as n goes to infinity but meant 'limit inferior' which is defined to be the infimum of all subsequential limits of the original sequence. I attach the link here.
Limit superior and limit inferior - Wikipedia, the free encyclopedia
So even in the case of 2+(-1)^n, lim inf abs{a(n+1)/a(n)} ≥ 1 still holds. In fact, it equals one in this case.