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

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

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,

$\sum _{n=0}^{\infty }\left({\frac {1}{2}}\right)^{n}=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots ={\frac {1}{1-{\frac {1}{2}}}}=2$