Geometric series

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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\ne 1 \\ a n & \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 xk 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.

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