Well,everypower serieshasa 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).

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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,hasa "radius of convergence" but it is0! The series converges for z= 0 but no other z.