It is a generalization of Taylor series. See @olver86 page 30.
In the context of the note exponential map#Motivation, we have that a vector field $X$ on a manifold $M$ gives rise to a flow in the sense that for a point $p\in M$ and a fixed "time" $t$ we obtain a point $q$ given by
$$ q=pe^{tX}=p\lim_{n\to \infty}(Id+tX/n)^{n}. $$The velocities of this curves are precisely $X$.
Now, given a smooth function $F$ defined on $M$, we observe that if $X=\sum \xi_i\partial_{x_i}$
$$ \dfrac{d}{dt}F(pe^{tX})=\sum_{i=1}^n \partial_{x_i}F(pe^{tX})\cdot \xi_i(pe^{tX}) $$and for $t=0$
$$ \left.\dfrac{d}{dt}\right|_{t=0} F(pe^{tX})=\sum_{i=1}^n \partial_{x_i}F(p)\cdot \xi_i(p)=X(F)(p). $$This can be written in another way, according to Taylor
$$ F(pe^{tX})\approx F(p)+tX(F)(p) $$ and, moreover, $$ F(pe^{tX})\approx F(p)+tX(F)(p)+\frac{t^2}{2}X^2(F)(p)+\frac{t^3}{3!}X^3(F)(p)+\cdots $$where $X^k(F)=X(X(\cdots(F)))$.
This expression is called Lie series.
If we take $F=(x_1,\ldots,x_n)$ the coordinate functions, then
$$ pe^{tX}=p+t\xi(p)+\frac{t^2}{2}X(\xi)(p)+\frac{t^3}{3!}X(X(\xi))(p)+\cdots \tag{*} $$where $\xi=(\xi_1,\ldots,\xi_n)$ and $X(\xi)=(X(\xi_1),\ldots,X(\xi_n))$ and so on.
In this sense, expression $(*)$ is a formal power series solution to the system of first order ODEs
$$ \dot{x}=X(x). $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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