A geometric sequence is a sequence in which each term is obtained from the last by multiplying by a fixed quantity, known as the common ratio. So for example, $$1, 2, 4, 8, 16, \dots$$ is a geometric sequence with common ratio 2, and $$81, -54, 36, -24, 16,\dots$$ is a geometric sequence with common ratio -2/3.
A geometric series is obtained by adding successive terms of a geometric sequence, as in $$81-54+36-24+16+\dots.$$ If the first term of a geometric sequence is a and the common ratio is r, then:
etc. In general, the nth term is ar^{n-1}.
If we sum the first n terms of a geometric series, we obtain $$S_n=a+ar+\dots+ar^{n-2}+ar^{n-1}.$$ Multiplying this sum by r, we get $$r S_n=ar+ar^2+\dots+ar^{n-1}+ar^{n}.$$ Subtracting the former from the latter gives $$S_n(r-1)=-a+(ar-ar)+(ar^2-ar^2)+\dots+(ar^{n-1}-ar^{n-1})+ar^n,$$ and thus $$S_n=a\frac{r^n-1}{r-1}.$$ An alternative form for this formula is $$S_n=a\frac{1-r^n}{1-r}.$$ If |r|<1, and only if this is the case, then r^n approaches 0 as n gets bigger, meaning that $$a\frac{1-r^n}{1-r}$$ approaches $$\frac{a}{1-r}.$$ This is known as the sum to infinity, S_{\infty}, of the series.
Notice that a geometric series has a sum to infinity if and only if its common ratio lies strictly between -1 and 1.