I would like to know if the following equality is always true for geometric series.
$$\sum_{k=n}^{\infty} q^{k} = \sum_{k=0}^{\infty} q^{n} \cdot q^{k}$$
It has worked for me on several occasions, for example:
$$ \sum_{k=1}^{\infty} 45 \cdot \left(\frac{25}{36}\right)^{k} = \sum_{k=0}^{\infty} 45 \cdot \left(\frac{25}{36}\right)^{1} \cdot \left(\frac{25}{36}\right)^{k} $$
$$ \sum_{k=2}^{\infty} 45 \cdot \left(\frac{25}{36}\right)^{k} = \sum_{k=0}^{\infty} 45 \cdot \left(\frac{25}{36}\right)^{2} \cdot \left(\frac{25}{36}\right)^{k} $$
So, can lower limits be safely shifted with this method?
Thanks!