is there a name for a series wich alternates + and - terms? Is there a specific name for a geometric series such as this?
$1-\frac{c (os\theta)}{k}+\frac{c^2}{k^2}-\frac{c^3}{k^3}+\frac{c^4}{k^4}-....$
How can we identify it if positive terms are even or odd?
Also, does its definition change if  the numerator follows a more complex pattern such as:
$1-\frac{c (os\theta)}{k}+\frac{c^2}{k^2}-\frac{c^3+c^2}{k^3}+\frac{c^4+c^3}{k^4}-....$

It's the Taylor expansion of $\frac 1 {1+ \frac c k}$
$$1-\frac{c}{k}+\frac{c^2}{k^2}-\frac{c^3}{k^3}+\frac{c^4}{k^4}-....=\sum_{n=0}^{\infty}(-1)^n( \frac c k)^n=\sum_{n=0}^{\infty}( \frac {-c} k)^n=\frac 1 {1+ \frac c k}$$
