Show that $Pr[X \gg Y]\approx 1$

Question

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ where one can choose $k \gg 1$ arbitrarily.

Answer 1

Let $b_n=\text{Bin}(n,n/2^n)$. Then $E(b_n)=n^2/2^2$, $\text{Var}(b_n)=(1-n/2^n)n^2/2^n$, and the series $\sum b_n$ converges in $L_2$. This means that $Y_k=\sum_{k+1}^\infty b_n$ converges to $0$ - and thus is smaller than $X$ most of the time.