Here is an interesting numerical means to compute the radius of convergence. It requires that the coefficients of x^n , alternate or are all the same sign.
We need to compute the ordered pairs:
The above generates:
Extrapolate the value in the second column for n = ∞. Looks like we get -.5625.
Here is another one:
This one is a little tougher we must extrapolate to 1 / ∞. So you do a linear fit of the last two points. You get:
The limit of that as x approaches 0 is obviously .2498259
Which is close to the the exact answer of 2.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
A number by itself is useful, but it is far more useful to know how accurate or certain that number is.