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.

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

$S_{n}=\sum _{k=0}^{n-1}x^{k}=1+x+x^{2}+\cdots +x^{n-1}={\begin{cases}{\displaystyle {\frac {1-x^{n}}{1-x}}}&{\hbox{for }}x\neq 1\\n\cdot 1&{\hbox{for }}x=1\end{cases}}$

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.