TITLE: Summation of odd power
QUESTION [8 upvotes]: How do I find the explicit formula for the following summation:
$x^1 + x^3 + x^5 + ... $
I know $1 + x + x^2 + x^3 + ...  = 1/(1-x)$, but this is quite a different series.

REPLY [18 votes]: $$x+x^3+x^5+....=x(1+x^2+x^4+....)=x(1+(x^2)^1+(x^2)^2+(x^2)^3+....)=\frac {x}{1-x^2}$$

REPLY [6 votes]: Not much different,$$x^1+x^3+x^5\cdots=x(1+x^2+(x^2)^2+\cdots)$$