Where is the mistake in the convolution?
$3^{rd}$ term in your final answer should have $6 \le n \le 10$, instead of $0 \le n \le 4$. And, the limit of summation will go from $n-4$ to $6$.
$$\sum^{6}_{k = n-4} \alpha^k \ ,  6 \le n \le 10$$
$0 \le n \le 4$ case has already been covered by the $1^{st}$ term of final answer.