TITLE: Formula for Alternating Geometric Series
QUESTION [5 upvotes]: I am aware of the following formula:
$$\sum_{n=0}^{\infty}(-1)^nr^n=\frac{1}{1+r}$$
However, I am having difficulty understanding if there is a simple formula for the following equation:
$$\sum_{n=0}^{\infty}(-1)^nr^{2n}={ }?$$
In addition, what about the following?
$$\sum_{n=0}^{\infty}(-1)^{n+1}r^{2n+1}={ }?$$
I'm not sure how to go about identifying how to modify the original formula to fit the previous two, if that is even possible. Any help would be greatly appreciated.

REPLY [9 votes]: $$\sum_{n=0}^{\infty}(-1)^nr^n=\sum_{n=0}^{\infty}(-r)^n=\cfrac{1}{1-(-r)}=\cfrac{1}{1+r}$$
So you can do the same for two following series :
$$\sum_{n=0}^{\infty}(-1)^nr^{2n}=\sum_{n=0}^{\infty} {(-r^2)}^n $$$$\sum_{n=0}^{\infty}(-1)^{n+1}r^{2n+1}=-r\sum_{n=0}^{\infty}(-1)^nr^{2n}$$