A real or complex, finite or infinite series a₁ + a₂ + a₃ + …in which the terms form a geometric sequence. Thus the terms have a common ratio r with a_k/a_k−1 = r for all k. If the first term a₁ equals a, then a_k = ar^k−1. Let s_n be the sum of the first n terms, so that s_n = a + ar + ar² + ⋯ + arⁿ⁻¹. Then s_n is given (when r ≠ 1) by the formulae

If the common ratio r satisfies |r|<1, then rⁿ → 0 and it can be seen that s_n → a/(1−r). The value a/(1−r) is called the sum to infinity of the series a + ar + ar² + …. In particular, for |x|<1, the geometric series 1 + x + x² + …has sum to infinity equal to 1/(1−x). For example, putting $x=\frac{1}{2}$, the series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$ as sum 2. If |x| ≥ 1, then s_n does not tend to a limit and the series has no sum to infinity.