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For ratio test and nth root test :

If the limit is more than 1 , then we say that the series diverges .

If the limit is less than 1 , then we say that the series converges.

If the limit is 1 , we say that the test fails .

My questions :

1- What if the limit is plus infinity ? Does the series diverges since infinity is more than 1 ?

2- What if the limit is minus infinity ? Does the series converges since minus infinity is less than 1 ?

3- What if the limit does not exist ?

MCS
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    For the first two cases it diverges trivially: the general term doesn't even tend to $0$. – Bernard Nov 26 '17 at 01:26
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    For both the ratio and the root test you should be taking absolute values, so the limit is always nonnegative. $\infty$ is bigger than $1$ so the series diverges in this case. If the limit doesn't exist then both tests are inconclusive; consider, for example, the series obtained by alternating $\frac{1}{2^n}$ and $\frac{1}{3^n}$. – Qiaochu Yuan Nov 26 '17 at 01:26

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