Solve for locus of points in the complex plane and please describe all sol'n steps

|z|= |z-i| z = x + yi

Find locus of all points in the complex plane.

Please give details of soln |z-i| + |z| = 9

Determine expressions from a given formulae

I have the answers to this question, although I would like to know how to get to answers from the given formulae. Please list all steps in as much detail as possible. thanks

Root test

a) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity" ps. {An} is a sequence of non-zero complex numbers b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle

Power series - holomorphic function

b) Show that for |z|<1 we have (z/1-z^2)+(z^2/1-z^4)+.......(z^2n/1-z^2n+1)=(z/1-z) and (z/1+z)+(2z^2/1+z^2)+....(2^kz^2k/1+z^2k)=(z/1-z) ps. maybe dyadic expansion of an integer may be used here or the fact that 1+2+2^2....+2^k=2^k+1 -1 c) Find the holomorphic function of z that vanishes at z=0 and has real part u(x,y)=(x(1+x^2+y^2))/(1+2x^2-2y^2+(x^2+y^2)^2)