Chapter 3.7, Problem 17E

The Taylor method of order $2$ can be used to approximate the solution to the initial value problem

$y^{\prime }=y$, $y\left(0\right)=1$,

at $x=1$ . Show that the approximation $y_n$ obtained by using the Taylor method of order $2$ with the step size $1/n$ is given by the formula

$y_n={\left(1+\frac{1}{n}+\frac{1}{2n^2}\right)}^n$, $n=1,2,\mathrm{...}$

The solution to the initial value problem is $y=e^x$, so $y_n$ is an approximation to the constant $e$.