how to convert /understand /reason the summation conversion? How to interpret / reason / understand the summation conversion for the below equation ?
$$\sum_{m = 1}^\infty 2^{-2m} = \frac{1}{4}\sum_{m = 0}^\infty 4^{-m}$$

There are two steps here. First, pulling down the $-2$ in the exponent:
$$\sum_{m=1}^\infty 2^{-2m} = \sum_{m=1}^\infty (\frac{1}{4})^m$$
Next, shifting the lower index to zero (by replacing $m$ everywhere with $m+1$) and pulling out a factor of $\frac{1}{4}$:
$$\sum_{m=1}^\infty (\frac{1}{4})^m = \sum_{m=0}^\infty (\frac{1}{4})^{m+1}
=\frac{1}{4}\sum_{m=0}^\infty (\frac{1}{4})^m=\frac{1}{4}\sum_{m=0}^\infty 4^{-m}$$
