The formula for the Taylor polynomial of degree `n` centered at `x_0` , approximating a function `f(x)` possessing `n` derivatives at `x_0` , is given by
`p_n(x)=sum_(j=0)^n (f^((j))(x_0))/(j!) (x-x_0)^j`
For `f(x)=e^x, f^((j))(2)=e^2` for all `j` .
Therefore to fourth order in `x` about the point `x_0=2`
Notice the graph below. The function `p_4(x)~~e^x` (red) the most at `x=2` . It will become more and more approximate to `e^x` the higher order the approximation.