Proof of Sum of Geometric Series

Below is the proof of sum of geometric series of sum of geometric progression.

The sum of geometric series

The sum of geometric series can be found from the difference between the geometric progression and some other progression which is a multiple of the progression. Let us define the following term for the proof of sum of geometric progression or geometric series.

And, for the other geometric series, let's multiply the above geometric series by r. This gives:

Next we subtract the second geometric series from the first geometric series. This gives: