Sum of an odd recursive sequence

Question

Let

$a_0 = 1$

$a_1 = 1 - \frac{e}{2}$

$a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$.

Find $\sum_{r=0}^{\infty}a_r$.