I was looking at how to derive the autocorrelation function of a stationary AR(1) process and ran into the following equality in one derivation I found:
$$ \sum_{i=0}^{\infty}\sum_{j=\tau}^{\infty}\phi^{i+j-\tau}=\sum_{j=\tau}^{\infty}\phi^{2j-\tau} $$ I'm having a hard time understanding how this equality is derived. Clearly if you separate $\phi$ into $\phi^i\phi^{j-\tau}$ and calculate the geometric sums separately you do not get the same result as what you get here ($\frac{\phi^\tau}{1-\phi^2}$).
The given equality is false.
$$\sum_{i=0}^\infty \sum_{j=\tau}^\infty \phi^{i+j-\tau} = \sum_{i=0}^\infty \phi^i \sum_{j=\tau}^\infty \phi^{j-\tau} = \sum_{i=\tau}^\infty \phi^{i-\tau} \sum_{j=\tau}^\infty \phi^{j-\tau} = \left(\sum_{j=\tau}^\infty \phi^{j-\tau}\right)^2$$