How is this the jet of $e^{sin t}$? Please could someone help me understand the following statement:
We can work out $k$-jets of composites by 'neglecting higher terms'. For example, the $4$-jet of $e^{\sin t}$ at $0$ is obtained from 
$$ 1 + (t - {1\over 6}t^3) + {1\over 2}(t - {1\over 6}t^3)^2 + {1\over 6}(t - {1\over 6}t^3)^3 + {1\over 24}(t - {1\over 6}t^3)^4$$
I see that this is the first few terms of the power series of $e^x$ evaluated at $t - {1\over 6}t^3$. 

Why does evaluating the series of $e^x$ at $t - {1\over 6}t^3$ yield
  the series of $e^{\sin t}$?


Well you only want the $4$-jet, so you can throw away any terms in the Taylor series of $\sin t$ of degree higher than $4$ since this these won't contribute to the $4$-jet of $e^{\sin t}\,,$ so you just use $t-\frac{t^3}{6}\,.$ However it's not true as you wrote that evaluating the series of $e^x$ at $t−\frac{t^3}{6}$ yields the series of $e^{\sin t}\,,$ if by series you mean Taylor series.
