I have tried to prove the equation:
(∇β∇νV)μ=Vμ;ν,β+Vα;νΓμαβ−Vμ;αΓανβ,
as found in Bernard Schutz book second edition "A first course in general relativity" in the paragraph 6.5 "The curvature tensor" and I found it's correct (here the link to my attempt to prove it: https://feelideas.com/riemann-curvature-tensor-and-covariant-derivative/).
But if I try to develop it in a different way I find a different result:
(∇β∇ν→V)μ=dxμ(∇β(Vα;ν∂∂xα))=dxμ(Vα;ν,β∂∂xα+Vα;ν∇β∂∂xα)=
dxμ(Vα;ν,β∂∂xα+Vα;νΓσαβ∂∂xσ)=dxμ(Vα;ν,β∂∂xα+Vσ;νΓασβ∂∂xα)=
dxμ((Vα;ν,β+Vσ;νΓασβ)∂∂xα)=Vμ;ν,β+Vσ;νΓμσβ
They differ for the term:
−Vμ;αΓανβ
Where am I wrong?