could someone please explaine to me how his power series has a radus of convergence and what it is:
the sum between n=0 and infinity of (n!)^2 Z^n. when Z member of the complex plain
Well, every power series has a radius of convergence!
To find the radius of convergence of , use the "ratio test" (that or, occaisionally the root test, is almost always the best way to find radii of convergence).
In order for the power series to converge that must go to a number less than 1 as n goes to infinity. In fact, it goes to infinity for all non-zero z but for z= 0, it is just 0.
That is, this series, like every power series, has a "radius of convergence" but it is 0! The series converges for z= 0 but no other z.