I don't understand two parts in the formula. In the first place how can we remove the sommation? Finally, in the last part how are the beta zero and the phi before the beta zero handled? Thank you in advance
$$\beta_0+\sum_{j=1}^\infty\phi^{j-1}(\phi\beta_0+\beta_1)=\beta_0+\frac{\phi\beta_0+\beta_1}{1-\phi}=\frac{\beta_0+\beta_1}{1-\phi}.$$
Since\begin{align}\sum_{j=1}^N\phi^{j-1}(\phi\beta_0+\beta_1)&=\beta_0\sum_{j=1}^N\phi^j+\beta_1\sum_{j=1}^N\phi^{j-1}\\&=\beta_0\frac{\phi-\phi^{N+1}}{1-\phi}+\beta_1\frac{1-\phi^N}{1-\phi},\end{align}we have\begin{align}\sum_{j=1}^\infty\phi^{j-1}(\phi\beta_0+\beta_1)&=\beta_0\frac\phi{1-\phi}+\frac{\beta_1}{1-\phi}\\&=\frac{\beta_0\phi+\beta_1}{1-\phi}.\end{align}It should be clear that$$\beta_0+\frac{\beta_0\phi+\beta_1}{1-\phi}=\frac{\beta_0+\beta_1}{1-\phi}.$$