$k$th power of ideal of of germs Well,We denote set of germs at $m$ by $\bar{F_m}$ A germ $f$ has a well defined value $f(m)$ at m namely the value at $m$ of any representative of the germ. Let $F_m\subseteq \bar{F_m}$ be the set of germs which vanish at $m$. Then $F_m$ is an ideal of $\bar{F_m}$ and let $F_m^k$ denotes its $k$th power. Could any one tell me how the elements look like in this Ideal $F_m^k$? and they said all finite linear combination of $k-fold$ products of elements of $F_m$ But I dont get this. and they also said These forms $\bar{F_m}\supsetneq F_m\supsetneq F_m^2\supsetneq\dots \supsetneq$

If $x_1,\ldots,x_n$ are local coordinates at the point $m$, then any smooth germ at $m$ has an associated Taylor series in the coordiantes $x_i$.  The power $F_m^k$ is precisely the set of germs whose degree $k$ Taylor polynomial vanishes, i.e. whose Taylor series has no non-zero terms of degree $\leq k$.
