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 $e^{\sin t}$?


Since
$$
\sin t = \sum_{k=0}^{\infty} (-1)^{k} \frac{t^{2k+1}}{(2k+1)!}
  = t - \frac{t^{3}}{3!} + \frac{t^{5}}{5!} - \cdots,
$$
you have $\sin t = t - \frac{t^{3}}{6}$ up to order four.
