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 is series expansion of alternating ( not specifically mentioned as alternating sign but it is implied) geometric series for
$$ \dfrac{1}{1+\dfrac{c}{k}}$$
Alternating series.. this refers to sign changes only regardless of numerator or denominator pattern in $n^{th}$ general term $a_n$. The changes are anyhow taken care of in the definition of $a_n$.
It's the Taylor expansion of $\frac 1 {1+ \frac c k}$
$$1-\frac{c}{k}+\frac{c^2}{k^2}-\frac{c^3}{k^3}+\frac{c^4}{k^4}-....=\sum_{n=0}^{\infty}(-1)^n( \frac c k)^n=\sum_{n=0}^{\infty}( \frac {-c} k)^n=\frac 1 {1+ \frac c k}$$
An alternating series is such that the values of the successive terms summed change in sign every time.
For example
$$1-1+1-1+1-1+1-1+\cdots,$$
but not
$$1-1-1+1-1-1+1-1+1+\cdots$$
In your first example, the series is alternating when $\dfrac ck>0$.
Your second example is a little harder. The first term is positive, hence the second must be negative, requiring $\dfrac ck>0$. Then the third is fine. The fourth must be negative and
$$\frac{c^3+c^2}{k^3}=\frac{c^2(c+1)}{k^3}>0.$$
Continuing with the next terms (assuming a regular pattern), the constraints are
$$\frac ck>0,\\\frac{c+1}k>0.$$
Hence,
$$(k<0\land c<-1)\lor (k>0\land c>0).$$