In calculus, we learn that e $=$ limh$0(1+$ h$)1/$h

$.$

a$.$ Determine approximations to e corresponding to h$=0\mathrm{.}04,0\mathrm{.}02,$ and $0\mathrm{.}01.$

b$.$ Use extrapolation on the approximations, assuming that constants K$1,$ K$2,...,$ exist with e $=$

$(1+$h$)1/$h $+$K$1$h$+$K$2$h$2+$K$3$h$3+,$ to produce an O$($h$3)$ approximation to e$,$ where h$=0\mathrm{.}04.$

c$.$ Do you think that the assumption in part $($b$)$ is correct?