Using central limit theorem , evaluate

Question

$\lim_{n\to \infty}\sum_{j=0}^{n}{j+n-1 \choose j}(\frac{1}{2^{n+j}})$,

I multiplied and divided the series by $1/2$ , And made it look like a binomial distribution ,but they are not i.i.d., which is why I cannot apply CLT.

Please help

Answer 1

Hint 1: In this problem you are considering the sum $S_n$ of $n$ i.i.d. random variables of geometrical distribution with parameter $1/2$. You are asking for the probability of $S_n\leq 2n$ and you may indeed apply the C.L.T.

Hint 2: In order to calculate the limit you may notice that the expectation of $S_n$ is precisely $2n$.