Geometric series

A geometric series is a series associated with an infinite geometric sequence, i.e., the quotient q of two consecutive terms is the same for each pair.

A geometric series converges if and only if −1<q<1.

A geometric series consisting of n terms is,

$a(1+x+x^{2}+\cdots +x^{n-1})\equiv a\sum _{k=1}^{n}x^{k-1},$

where a and x are real numbers. It can be shown that

$S_{n}\,{\stackrel {\mathrm {def} }{=}}\,a\sum _{k=1}^{n}x^{k-1}={\begin{cases}{\displaystyle a{\frac {1-x^{n}}{1-x}}}&{\hbox{for}}\quad x\neq 1\\an&{\hbox{for}}\quad x=1\end{cases}}$

The infinite geometric series $a\sum _{k=1}^{\infty }x^{k-1}$ converges when |x| < 1, because in that case x^k tends to zero for $k\rightarrow \infty$ and hence

$\lim _{n\rightarrow \infty }S_{n}={\frac {a}{1-x}},\quad {\hbox{for}}\quad |x|<1.$

The geometric series diverges for |x| ≥ 1.