Geometric series

The geometric series are a family of series of the form

\infty \sum 𝑛 = 0 𝑟 𝑛

such series converge if and only if $| 𝑟 | < 1$ , and in these cases the result is determined to be

𝑠 = \infty \sum 𝑛 = 0 𝑟 𝑛 = 1 1 - 𝑟 .

This follows from the partial sum (for $𝑟 \neq 1$ ) is given by

𝑠 𝑘 = 𝑘 \sum 𝑛 = 0 𝑟 𝑛 = 1 - 𝑟 𝑘 + 1 1 - 𝑟