Does the sum
$$
\lim_{n\to\infty}\sum_{k=0}^{\lfloor\alpha n \rfloor}C_n^k(-1)^k\left(1-\frac{k}{\alpha n}\right)
$$
converge, where $C_n^k$ is the  binomial coefficient  and $0 <\alpha <1$?


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The above question has been [solved](https://mathoverflow.net/a/366433) by Iosif Pinelis. A variation is
$$
\lim_{n\to \infty}\sum_{k=0}^{\lfloor\alpha n \rfloor}C_n^k(-1)^k\left(1-\frac{k}{\alpha n}\right)^n.
$$
How can we handle this sum?