Showing an asymptotic lower bound

Question

How can we show that $$1 - 3(1-2/x)^y + 2(1-3/x)^y = \Omega((y/x)^2)$$?

We probably want to use a Taylor expansion here, but we have two variables.

Answer 1

Working the general term $$A_k=\left(1-\frac{k}{x}\right)^y\implies \log(A)=y \log\left(1-\frac{k}{x}\right)$$ $$\log(A)=y\left(-\frac{k}{x}-\frac{k^2}{2 x^2}-\frac{k^3}{3 x^3}+O\left(\frac{1}{x^4}\right) \right)$$ $$A_k=e^{\log(A_k)}=1-\frac{k y}{x}+\frac{k^2 (y-1) y}{2 x^2}-\frac{k^3 (y-2) (y-1) y}{6 x^3}+O\left(\frac{1}{x^4}\right) $$ $$1-3A_2+2A_3=\frac{3 y^2-3 y}{x^2}+\frac{-5 y^3+15 y^2-10 y}{x^3}+O\left(\frac{1}{x^4}\right) $$ So, if $y$ is large, pushing the expansion of $A_k$, $$1-3A_2+2A_3\sim 3 \left(\frac{y}{x}\right)^2-5\left(\frac{y}{x}\right)^3+\frac{19}4 \left(\frac{y}{x}\right)^4+\frac{13}4 \left(\frac{y}{x}\right)^5+\cdots$$