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First, we use the ratio test. Since we have
when the series converges, while when the series diverges.
Since is equivalent with and , we can rephrase that the series converges for and diverges for and .
Now we consider the endpoints; when . Plugging this into the summation we have
Then, we can observe that the sequence doesn't converges to 0 as goes to infinity.
Indeed, if we assume that it converges to 0, by limit rule we have the convergence of the sequence at infinity:
However, apparently, this is not true when we expand the sequence :
Therefore, by the divergence test, the given series at diverges.
On the other hand, the given series at also diverges. This follows from again the divergence test. Indeed, at the series can be written as
Since the sequence doesn't converges to 0 as goes to infinity, so is .
Collecting all the information, the given series converges for .