Interactive Real Analysis

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Ratio Test

Examples 4.2.16(c):

The following statements are not equivalent:
• There exists an N such that | an+1 / an | 1 for all n > N
• lim sup | an+1 / an | 1
In fact, the first statement implies the second, but not the other way around.
Suppose that the first statement is true, i.e.
There exists an N such that | a n+1 / a n | 1 for all n > N
Now recall the definition of the lim sup as the limit of the supremums of the truncated sequences:
lim sup | a n+1 / a n | = lim ( sup{ | a n+1 / a n | , j n } )
But if n > N, then the expression | a n+1 / a n | 1. Therefore, the 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

2, 1/2, 2, 1/2, 2, 1/2, ...
Here the 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
| a n+1 / a n |
So, let
• a n = 2 if n is even
• a n = 1 if n is odd
Then
• a n+1 / a n = 1 / 2 if n is even
• a n+1 / a n = 2 / 1 if n is odd
Therefore, the second statement above does not imply the first one.
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