I intended to use facts specifically about the radius of convergence of a power series instead of tests that apply to any series.
(set k = n + 1, i.e., n = k - 1)
(set n = 2k, i.e., k = n/2)
where
Now, strictly speaking, we cannot use the formula
because some are equal to 0 (but see Soroban's answer on how to use ratio test for arbitrary series), but we can use
Since as , , so r = 1/2. If we did not convert the series to the form and took the nth root of , we would get r = 1/4.