A geometric sequence is a sequence in the form

${\displaystyle \{a_n\}_{n=0}^\infty=ar^n}$

Where r is a common ratio. If ${\displaystyle |r|<1}$ , the sequence is convergent, and has a defined limit. If ${\displaystyle |r|>1}$ , the sequence is divergent, and will have an undefined limit. An example of a convergent sequence is

${\displaystyle \{a_n\}_{n=0}^\infty=8\left(\frac12\right)^n=8+4+2+1+\frac12+\frac14+\cdots}$

which will converge to 0.

A geometric series is the sum of a geometric sequence. The sum of the series is equal to

${\displaystyle \sum_{k=0}^n ar^k=\frac{a(r^{n+1}-1)}{r-1}}$

If a geometric series is infinite and convergent, the formula will still apply (again, if ${\displaystyle |r|<1}$ , the series is convergent), but since

${\displaystyle \lim_{n\to\infty}\frac{a(r^{n+1}-1)}{r-1}=\frac{a}{1-r}}$

we can use the latter formula. For example,

${\displaystyle \sum_{n=0}^\infty\left(\frac12\right)^n=1+\frac12+\frac14+\frac18+\cdots=\frac{1}{1-\frac12}=2}$