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 &minus;1<q<1.

Its sum is $$ a \over 1-q $$ where a is the first term of series.

Power series
Any geometric series
 * $$ \sum_{k=1}^\infty a_k $$

can be written as
 * $$ a \sum_{k=0}^\infty x^k $$

where
 * $$ a = a_1 \qquad \textrm{and} \qquad x = { a_{k+1} \over a_k } \in \mathbb C

\hbox{ is the constant quotient} $$

The partial sums of the power series are

S_n = \sum_{k=0}^{n-1} x^k = 1 + x + x^2 + \cdots + x^{n-1} = \begin{cases} {\displaystyle \frac{1-x^n}{1-x}} &\hbox{for } x\ne 1 \\ n \cdot 1 &\hbox{for } x = 1 \end{cases} $$ because
 * $$ (1-x)(1 + x + x^2 + \cdots + x^{n-1}) = 1-x^n $$

Since
 * $$ \lim_{n\to\infty} {1-x^n \over 1-x } = {1-\lim_{n\to\infty}x^n \over 1-x } \quad (x\ne1)$$

there is
 * $$ \lim_{n\to\infty} S_n = {1 \over1-x } \quad \Leftrightarrow \quad |x|<1 $$

and the geometric series converges for |x|<1 with the sum
 * $$ \sum_{k=1}^\infty a_k = { a \over 1-q }$$

and diverges for |x| &ge; 1.