Taylor expansion of matrix square root about A which is not the identity matrix.

Question

In https://physics.stackexchange.com/questions/196448/square-root-of-a-matrix-appears-in-massive-gravity-how-to-solve-sqrtab-pe, there is a claim $\sqrt{A+B}=\sqrt{A}+\sum_{n=0}^\infty\binom{\frac{1}{2}}{n}\left(\left(L_B+adA \right)^{n-1}B\right)A^{\frac{1}{2}-n}$ for $A>0$ and $B$ sufficiently small. $L_B(A):=BA$ and $(adA)B:=[A,B]$. Can somebody please help me with finding the source of this claim, or how to prove it?

Answer 1

Hint: $$ \sqrt{A+B} = \sqrt[4] A\sqrt{I + A^{-1/2}BA^{-1/2}}\sqrt[4] A $$