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Proof

# Sum of a Geometric Series

What follows is a derivation of the formula for the sum of a geometric series.
Consider a geometric series of terms, whose first term is and whose common ratio is The sum of the series can be written as
Notice that the series is a polynomial, and can be written in standard form.
Since all of the terms contain a factor of it can be factored out of the right-hand side.
Since the sum above is a polynomial, polynomial division can be used to rewrite it. In fact, one polynomial identity gives
Thus, the formula for the sum of a geometric series can be written as follows.

Q.E.D.