Long run multiplier equality I don't understand the following equality:
$$ \beta_0 +  \sum_{j=1}^\infty \phi^{j-1} (\phi\beta_0 + \beta_1) = \beta_0 + \frac{\phi\beta_0 + \beta_1}{1 - \phi} $$
It calculates the long run multiplier from an ADL(1,1) model.
Could someone explain it or maybe show some steps in between?
Thanks

Hint. If $|x|<1$ then the geometric sum evaluation leads to
$$
\sum_{j=1}^\infty x^{j-1}=\frac1{1-x}
$$ giving
$$
\sum_{j=1}^\infty \phi^{j-1}=\frac1{1-\phi},\qquad |\phi|<1.
$$
