Could you check if these two are equivalent?

  • #1
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.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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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 "[itex]\ge[/itex]" 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 [itex]\ge[/itex] with ">" in each.

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

Thank you in advance.
 
  • #3
34
0
If it equals one, the ratio test is inconclusive.
 

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