I am working on a homework problem and I was wondering if someone could possibly give me a hint (or a solution) on how to show that the sum $\sum_{k\geq 1} (1- Q_k)$ diverges, where $$ Q_k = \sum_{0 \leq j \leq \lfloor 2^k/k \rfloor } {2^k-kj \choose j} (-p^{k}(1-p))^j - p^k\sum_{0 \leq j \leq \lfloor 2^k/k - 1 \rfloor } {2^k-kj-k \choose j} (-p^{k}(1-p))^j, $$ and $p \geq 1/2$.
I would also like to note that $1-Q_k$ is what I computed as the probability that a {0,1} string of length 2^k contains a 0-block of length greater than or equal to k if the occurrence of 0 has probability p and 1 has probability 1-p.
Thank you very much!