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(1)Consider the series ∑z^n,|z|<1 z is in C

I thik this series is absolutely and uniformly comvergent since the series ∑|z|^n is con vergent for |z|<1,but I have a book saying that it is absolutely convergent,not uniformly.........i am confused...

(2)for the function f(z)=1/√(z-1),z=1 is a (a)pole (b)an essential singularity ?

I think it is an essential singularity since if it is a pole,say of order m then m is a positive integer and we can write f(z)=g(z)/(z-1)^m, where g(z) is analytic at z=1 and g(1)≠0,

but the given function cannot be written in this way,but the answer is given pole,i am again confused...

Can anybody help me?