A general term for this series is+-x^r, having a positive sign when r is even and a negative sign when r is odd.

​​​​Now because (-1) is positive when r is even and negative when r is odd, the general term can be written as (-1)^rx^r

Hence;

1-x+x²-x³+...=£^o_r=0(-1)^r(x)^r