using complex conjugate properties and this fact that iff we get:

thus is holomorphic too and

Find the radii of convergence of the following series stating which result is being used.

(a)

(b)

(c)

(A)

Using Ratio Test:

=

So is rad. of convergence (z-1)/2 and converging since it is less than 1??[/tex]

you should say that rather than putting !!!

so you got which means that the radius of convergence is 2.

(B)

Using Ratio Test:

Hence, is ROC infinity and diverging since it is greater than 1?

so the series is divergent for all values of z except for thus the radius of convergence is 0.

(C)

Using Ratio Test:

So is ROC 0 and converging since it is less than 1?

where did come from??! since you'll get:

so the radius of convergence is