Autoregressive Process of order $2$: long-run impact.

Question

Suppose we have Autoregressive Process of order $2$ abbreviated as: $AR(2)$

$$ y_t = \alpha_1 \times y_{t-1} + \alpha_2 \times y_{t-2} + \epsilon_t $$

Suppose at time $ t=2 $, $ \epsilon_0=1 $ and then again and before $ \epsilon =0 $ $ \forall t $.

For the last two hours I am trying to calculate the long run effect of such change on y. In the Long Run effect I mean what happens to $ \sum_{t=0}^{\infty} y_t $, i.e. how does the sum changes by such change at time $ t= 0 $.

For the case of $ **AR(1)$**, when $ y_t = \alpha_1 \times y_{t-1} + \epsilon_t $ it is easy to calculate.

Suppose at time $ t=1 $, $ \epsilon_0=1 $ and then again and before $ \epsilon =0 $ $ \forall t $.

1. In period 1 $ y_1 $ increases by $ 1 $
2. In period 2 $ y_2 $ increases by $ \alpha_1 $
3. In Period 3 $ y_3 $ increases by $ \alpha_1^2 $
4. ...
5. In period t $ y_t$ increases by $\alpha_1^{t-1} $

So, in total, $ \sum_{t=0}^{\infty} y_t $ increases by $ \frac {1}{1-\alpha_1} $ provided that $ \alpha_1 < 1$.

I want to calculate the long run effect in the same fashion for the $ AR(2)$.

1. In period 2 $y_2 $ increases by $ 1 $
2. In period 3 $ y_3$ increases by $ \alpha_1 $
3. In period 4 $ y_4 $ increases by $ \alpha_1^2+\alpha_2 $
4. In period 5 $ y_5 $ increases by $ \alpha_1^3 + 2 \alpha_1 \alpha_2 $
5. In period 6 $ y_6 $ increases by $ \alpha_1^4 +3 \alpha_1^2 \alpha_2 + \alpha_2^2 $
6. In period 7 $ y_7 $ increases by $ \alpha_1^5 +4 \alpha_1^3 \alpha_2 + 3 \alpha_1 \alpha_2^2 $
7. In period 8 $y_8 $ increases by $ \alpha_1^6 + 5 \alpha_1^4 \alpha_2 + 6 \alpha_1^2 \alpha_2^2 + \alpha_2^3 $
8. and so on... I could not identify pattern for $ t $ in general. Neither I could not identify the sum those numbers.

I found in one paper that it in general, for $ AR(p) $ Long Run Effect is $ \frac{1}{1-\sum_{i=1}^{p} \alpha_i} $.

Can anyone help me out in derive this result? I am having hard time falling asleep without getting this result.