Last step of infinite series problem unclear

[RoughRoad]

On the provided attachment, I have problem understanding the last step of the 1st sum of that page. Can anyone explain to me what is being done after the second-last step?

 

[Citan Uzuki]

They're using the formula for the sum of a finite geometric series:
$$\sum_{k=1}^{n}r^k = r\frac{1-r^n}{1-r}$$

 

[RoughRoad]

How is it a finite series? Aren't the term going till infinity? Sorry but I am actually a litle weak on my basics.

 

[Citan Uzuki]

The term s_n is the sum of just the first n terms, which is a finite series. The value of the infinite series is obtained by taking the limit of s_n as n→∞.

 

[RoughRoad]

Thanks! :-)