Geometric sequence

A geometric sequence is a sequence in the form

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

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

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

which will converge to 0.

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

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

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

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

we can use the latter formula. For example,

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