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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.