Evaluating sum of binomial coefficients This sum popped out of one of my calculations, I know what it should evaluate to, but I have no idea how to prove it. 
$$\sum_{i=0}^{r}{n \choose 2i } - {n\choose 2i -1}$$
I know that $2i-1$ is negative for $i=0$, but for the purpose of this sum, we will say that  ${n \choose x}=0$ if $n<0$ or $x<0$. So this sum is basically summing the difference of consecutive even/odd binomial coefficient pairs. We can rewrite this sum as
$$\sum_{i=0}^{2r}(-1)^i {n \choose i}.$$
I don't really know how to proceed from here, I couldn't find any information on evaluating sums of alternating series involving binomial coefficients.

Consider the coefficient of $x^{m}$ in the expansion of
$$(1-x)^n\frac{1}{1-x}=\sum_{r=0}^{n}(-1)^r{n \choose r}x^r\sum_{r=0}^{∞}x^r$$
