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Series[1/2 (e1 + e2 - Sqrt[e1^2 - 2 e1 e2 + e2^2 + 4 V12^2]), {V12, 0, 2}] // Simplify[#, e1 > e2] &


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If we remove the subscripts from the converted TeX, ToExpression[ "\\frac{1}{2} \\left(e_1+e_2-\\sqrt{e_1^2-2e_1e_2+e_2^2+4V_{12}^2} \\right)", TeXForm] /. Subscript[b_, e_] :> b[e] (* 1/2 (e[1] + e[2] - Sqrt[e[1]^2 - 2 e[1] e[2] + e[2]^2 + 4 V[12]^2]) *) we can simply use PowerExpand: Series[ 1/2 (e[1] + e[2] - Sqrt[e[1]^2 - 2 e[1] e[2] + e[2]^2 ...


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Assumptions given to Series are not used by FullSimplify that you have in your code. To pass Asumptions to all functions in given expression, wrap said expression with Assuming: Assuming[e1 > e2, Series[1/2 (e1 + e2 - Sqrt[e1^2 - 2 e1 e2 + e2^2 + 4 V12^2] ), {V12, 0, 2}] // Simplify ] (* e2 + V12^2/(-e1+e2) + O[V12]^3*) Or, if you want to always ...



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