# What is the geometric power series?

Dec 7, 2015

$a + a x + a {x}^{2} + a {x}^{3} + \cdots$, which converges to $\frac{a}{1 - x}$ when $| x | < 1$. More generally, you could also write $a + a \left(x - c\right) + a {\left(x - c\right)}^{2} + a {\left(x - c\right)}^{3} + \cdots$, which converges to $\frac{a}{1 - \left(x - c\right)} = \frac{a}{\left(1 + c\right) - x}$ when $| x - c | < 1$.