$F_p$ is the set of germs of functions on a manifold M which vanish at $p \in M$. Let $F^k_p$ be the ideal of $C^\infty(p)$ generated by $f_1,... \,f_k$, where $f_i \in F_p$. (i.e $F^k_p$ is $\sum g_if_{i1}...f_{ik}, g_i \in \mathbb C^\infty(p), f_{ij} \in F_p.)$
Prove that, in every coordinate system $(x_1, . . . , x_n)$, an element $f \in F^k_{p}$ has a Taylor expansion which vanishes up to order k
Can you give me hints on how to approach the problem or maybe online lectures available about this topic so I can read, cause I think I don't understand it perfectly, that's why I don't know how to start
Thanks