Let $v$ be a smooth function on a manifold and $R$ denote the Riemann Curvature tensor. If we work in normal coordinates, i.e. Christoffel symbols are zero why is the following formula true
$\nabla_i \nabla_j \nabla_k v = \nabla_{j}\nabla_i \nabla_k v + R_{ijk\ell} v_{\ell}?$
I tried to plug this into the definition of the Riemann curvature tensor but I am bit confused about what happens to the vector field $\partial_k v$. Any comments/suggestions will be much appreciated.