INdexing in the sum of a geometric series in derivation of geometric dist cdf

Hi, I can't see how the indexing is working in this derivation. The question is: Show that the cdf for a geometric random variable is given by , where denotes the greatest integer in .

The derivation is given as

But

From which the result follows.

However, the sum of a the first n terms in a geometric series is:

So if we are saying that and

Then I can't see how this makes sense since it seems we are summing over terms, from to

Or in other words, if , why isn't the closed form of the sum over as follows: ?

Thanks in advance for any insights. MD