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[Side remark: Use the button with 101 and 010 to format your code.]
You simply need to ask for the taylor expansion in z at $0$:
sage: i,z = var('i,z')
sage: c = function('c')
sage: p = sum(c(i)*z^i,i,0,6)
sage: psqrt = sqrt(p)
sage: psqrt.taylor(z, 0, 3) # I take 3 to limit the size of the output...
1/2*z*c(1)/sqrt(c(0)) + 1/16*(c(1)^3*sqrt(c(0)) - 4*c(2)*c(1)*c(0)^(3/2) + 8*c(3)*c(0)^(5/2))*z^3/c(0)^3 - 1/8*(c(1)^2*sqrt(c(0)) - 4*c(2)*c(0)^(3/2))*z^2/c(0)^2 + sqrt(c(0))
