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
We also know that there is divergence for
It remains to individually check on the point