To be more specific, we want to evaluate the limit $\lim\limits_{n\to\infty} \sum_{k=0}^{\lfloor c\cdot \log(n)\rfloor} e^{-\lambda_n} \frac{\lambda_n^k}{k!}$ where $\lambda_n=\sum_{j=1}^{n}\frac{1}{j}$.
A helpful fact, I think, is that $\frac{\lambda_n}{\log(n)}\to 1$ as n goes to infinity. Intuitively speaking, it makes sense that when $c\geq 1$ the limit is 1 since the summation will "include" the largest summands. Whereas if $c < 1$ then the summands in the series will all approach $0$ since the number of summands is not "keeping up" with the growing rate parameter. Therefore the limit of the CDF in this case is $0$. I am basing my intuition in the idea that the largest summands are those where $k$ is very close to $\lambda_n$. I am unsure how to prove the result in a rigorous way. A probabilistic approach would be appreciated.