
Limits are a concept that goes back at least as far as Archimedes, a Greek scientist, engineer and mathematician in the 3rd century BCE. He calculated a value for π based on the limit of measurements external and internal to a circle.
The sum of a finite geometric series is: $S_n = {u_1(1-r^n)}/{1-r}$, where $r$ is the common ratio of two consecutive terms, and $n$ is the number of terms $u$.
For a geometric series, ${Σ}↙{n=0}↖{∞} = {lim}↙{n→∞} {u_1(1-r^n)}/{1-r}$.
When $-1 < r < 1$, ${lim}↙{n→∞} r^n = 0$, and the series converges to its sum, $S={u_1}/{1-r}$.
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