I know from power series for square root matrix that $U=(I+A)^{1/2}$ can be expanded to power series provided $||A||<1$. How can I expand $(B+A)^{1/2}$ when $B$ is Hermitian (edited) positive definite and $A=\bf{u}\bf{u}^H$ is a rank-1 perturbation (modification). That is, $\bf{u}$ is a vector. What is the condition for convergence?
The equivalent trick for $U=(B+A)^{-1}$, where $B$ is taken out, seems not to work in the case of square root.