Given: $\bar{V}_t = \sum\limits_{s=t}^\infty \left[\left(\frac{1}{1+R}\right)^{s-t} \cdot \Pi_s \right]$
Goal: Rewrite to $\bar{V}_0 = \sum\limits_{t=0}^\infty \left[\left(\frac{1}{1+R}\right)^t \cdot \Pi_t \right]$
My attempt: $\sum\limits_{s=t}^\infty \left[\left(\frac{1}{1+R}\right)^{s-t} \cdot \Pi_s \right] = \sum\limits_{s=0}^\infty \left[\left(\frac{1}{1+R}\right)^{(s+t)-t} \cdot \Pi_{s + t} \right] = \sum\limits_{s=0}^\infty \left[\left(\frac{1}{1+R}\right)^{s} \cdot \Pi_{s + t} \right]$ (index shift), or, equivalently: $\sum\limits_{s=t}^\infty \left[\left(\frac{1}{1+R}\right)^{s-t} \cdot \Pi_s \right] = \sum\limits_{\tilde{t}=0}^\infty \left[\left(\frac{1}{1+R}\right)^{(\tilde{t}+t)-t} \cdot \Pi_{\tilde{t}+t} \right] = \sum\limits_{\tilde{t}=0}^\infty \left[\left(\frac{1}{1+R}\right)^{\tilde{t}} \cdot \Pi_{\tilde{t}+t} \right]$ (index change), but this does not lead to the required result (yet).
What am I missing here?
You are close. You have $$\bar{V}_t = \sum\limits_{s=0}^\infty \left[\left(\frac{1}{1+R}\right)^{s} \cdot \Pi_{s + t} \right]$$ Now substitute $t=0$ on both sides: $$\bar{V}_0 = \sum\limits_{s=0}^\infty \left[\left(\frac{1}{1+R}\right)^{s} \cdot \Pi_s \right]$$ Now change $s$ to $t$: $$\bar{V}_0 = \sum\limits_{t=0}^\infty \left[\left(\frac{1}{1+R}\right)^{t} \cdot \Pi_t \right]$$