geometric series

Geometric series is the sum of a geometric sequence, in which each term is a constant factor $r$ (a.k.a. common ratio) multiple of the previous term.

As long as $|r|<1$ , we can obtain the actual value of the infinite geometric series, which is the sum for all terms in the sequence:

$$\sum _{k=0}^{\infty }a{r}^k=\frac{a}{1-r},|r|<1$$

We can also obtain partial geometric series, which is the sum of the sequence up until the $n$-th term:

$$\sum _{k=1}^{n}a\cdot r^k=a\cdot \frac{r(1-r^n)}{1-r},r\ne 1$$