Here we will try to apply the standard tests to determine the convergence of the series

We treat x as parameter and sum up with respect to k. What tests can be used? Comparison seems out, since there are no parts that could be ignored for k large. However, the power suggests that this series is just made for the Root test (after we put terms into absolute value so that they become non-negative).

Thus the given series converges absolutely whenever the ratio in absolute value is less than 1, which is the condition we already had in the "official solution" and we know that it is true for x > −1/2.

We also know that there is divergence for x < −1/2, here we use the version of the Root test for general series, see e.g. the note at the end of the section Root and Ratio tests in Theory - Testing convergence.

It remains to individually check on the point x = −1/2, but we already did that.