For formula_5, the sum of the first "n" terms of a geometric series is

where is the first term of the series, and is the common ratio. We can derive this formula as follows:
so, 
As goes to infinity, the absolute value of must be less than one for the series to converge. The sum then becomes

When , this can be simplified to

the left-hand side being a geometric series with common ratio .

The formula also holds for complex , with the corresponding restriction, the modulus of is strictly less than one.

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