Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

Euler's method with $n$ equal steps on $y' = y$, $y(0) = 1$, over $[0, 1]$ gives $E_n = \left(1 + \dfrac{1}{n}\right)^n$. Its error has an expansion $E_n = e - \dfrac{e}{2n} + \dfrac{C}{n^2} + O\!\left(\dfrac{1}{n^3}\right)$. Find $C$, i.e. $\displaystyle\lim_{n \to \infty} n^2\left(E_n - e + \dfrac{e}{2n}\right)$.