In problems 1-6, determine the convergence set of the given power series.
The set is .
Use The Ratio Test to find the radius of convergence. In this case,
Therefore, the radius of convergence is , which means that the series converges for or, that is.
(1): Remove the absolute value, since always.
(2): Strip out the largest power of in the denominator and then cancel it
(3): Here and .
Now, we need to check whether the boundary points and are also in the set. We do that, by substituting and in the series.
Substituting for gives:
This is an alternating series. Therefore, use the alternating series test, to check whether it converges. First, check if
In this case,and it follows that.
(Also, is not monotone decreasing, which is the other condition of the alternating series test.) Therefore, the series above diverges and is not in the set.
Substituting 0 for gives:
Here, use the fact that if a series converges, then
By contraposition, it follows that
In this case, that is:
Now, forit follows that
Therefore, the series above diverges and is not in the set
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