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Using the ratio test, we have that
Now the ratio test tells us that our original series converges when .
This occurs when so whenever the distance from 2 is bounded above by 3.
This occurs when . Another way to see this is to note that is equivalent to and subtracting two from each
side gives .
The ratio test also tells us that the series diverges when and this occurs when or .
Isolating for x yields or .
This leaves only the cases when , that is, when . To check these, we plug them into the original series. For we have
Let's look at the limit of the terms. Since as n tends to infinity, we have that the sequence defined by
has two subsequences that tend to different values (the even indexed terms limit to 1 since the numerator is positive and the odd terms limit to -1 since the numerator is negative). Therefore, the divergence test tells us that the series diverges when
When , we have that
Let's look at the limit of the terms. Since as n tends to infinity, we have that the sequence defined by
converges to 1 as n tends to infinity. Therefore, the divergence test tells us that the series diverges when .
Thus the set of x values where this series converges is defined by .
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