Could you check if these two are equivalent?

[Chsoviz0716]

Hi guys,

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 in the sequence, and abs means absolute value.

Thank you in advance.

 

[HallsofIvy]

Hi guys,

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.

No, it doesn't say that. You need to replace "$\ge$" with ">".

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

Yes, if you replace that $\ge$ with ">" in each.

*a(n) means nth term in the sequence, and abs means absolute value.

Thank you in advance.

 

[Benn]

If it equals one, the ratio test is inconclusive.