I came across a lecture on generating functions which claimed: $$\frac{1}{(1-x)^4}$$ to be equal to $$\sum_{k=0}^4 \binom{-2}{k} x^k $$ to be equal to
$$1 + \frac{2}{x}+ \frac{3}{x^2}+ \frac{4}{x^3}+ \frac{5}{x^4}$$
I find this quite puzzling; the only way I can reach the 3rd expression is via $$\sum_{k=0}^4 \binom{-2}{k} (-x)^{-k} $$ which is similar to the second expression but I cannot find anything similar to the first expression that is equal to the 3rd.
Is there a way to prove the equivalency of the first three expressions?
The video lecture, second from last identity.