I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance
I want to understand the covariant taylor series part (page 87). Let $f$ be a scalar function on a Riemannian manifold. We have the Taylor series
$$f(x(t))=\sum_{n=0}^{\infty}\frac{1}{n!}t^n\left[\frac{d^n}{d\tau^{n}}f(x(\tau))\right]_{\tau=0}$$ Here $x(t)$ describes a geodesic from $x(0)=x'$ and $x(t)=x$. Now we have that $\frac{d}{d\tau}=\dot{x}(\tau)\nabla_{i}$ and the geodesic equation $\dot{x}^{i}\nabla_{i}\dot{x}^{j}=0$ ($\nabla$ is the Levi Cevita connection). Then he claims $$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}t^n\dot{x}^{i_{1}}(0)...\dot{x}^{i_{n}}(0)[\nabla_{(i_{1}}...\nabla_{i_{n})}f](x')$$ The parantheses denote symmetrization. [...] denotes the value inside when $x\rightarrow x'$
Where does this symmetrization comes from? For me it is not necessary.