Alternating geometric series, not sure what i am looking at

Question

I have written out a series describing a model system but i cannot find if there is a historical representation of the like series:

$$N_s= N\sum_{i=1}^n(-1)^{i-1}P^i$$

It appears to be an alternating geometric series, similar to the Mercator series.

Answer 1

$$\sum_{i=1}^n(-1)^{i-1}P^i=-\sum_{i=1}^n(-P)^i=-\left(\frac{1-(-P)^{n+1}}{1-(-P)}-1\right)=\frac{P+(-P)^{n+1}}{1+P}$$

See here for details.

Then

$$N_s=N\frac{P+(-P)^{n+1}}{1+P}$$