A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number. The first term is usually denoted as "a" and the fixed number is denoted as "r". The general form of a geometric series is a, ar, ar^2, ar^3, ..., ar^n.
The formula for finding the sum of a geometric series up to infinity is S = a / (1-r), where "a" is the first term and "r" is the common ratio.
A geometric series will converge (have a finite sum) if the absolute value of the common ratio "r" is less than 1. If the absolute value of "r" is greater than or equal to 1, the series will diverge (have an infinite sum).
Yes, a geometric series can have a negative common ratio. This means that the series will alternate between positive and negative terms, but the overall sum may still converge or diverge depending on the value of "r".
The formula for finding the sum of a geometric series up to a certain number of terms (n) is Sn = a (1-r^n) / (1-r), where "a" is the first term and "r" is the common ratio. This formula can be used to find the partial sum of a geometric series up to any number of terms.