Finding conditions for $\lim\limits_{n\to\infty}\sum\limits_{k=0}^{\lfloor\frac1{a_n}\rfloor}(-1)^k\binom nk(1-ka_n)^{n-1}=1$ 
Question: Find necessary and sufficient condition on the sequence $(a_n)_{n=1}^∞$ so that$$\lim_{n→∞}\sum_{k=0}^{\lfloor\frac1{a_n}\rfloor}(-1)^k\binom nk(1-ka_n)^{n-1}=1\tag 1$$given that $\lim\limits_{n\to\infty}a_n=0$ and $a_n\gt 0$ for all $n\in\Bbb{N}$.

After some guesswork I got to a condition that if $\sum\limits_{n\ge 1} a_n=\infty$ then eq.(1) holds. But I was not able to prove it neither could I find a counterexample for the conjecture. Searching on internet I found that this sum is very closely related to a special case of Dvoretzky covering problem but still couldn't find the necessary and sufficient condition. Until now I have tried using approximations for the Binomial Coefficient and binomial approximation to tackle the sum to no avail. I would be glad if someone could help.
Edit: I have got a counterexample for my conjecture i.e. $\sum\limits_{n\ge 1} a_n=\infty$ is alone not sufficient for eq.(1) to hold. So what should be the necessary and sufficient condition?
 A: This isn't an answer so much as me reporting what I've found simply testing different $a_n$ sequences. First, if $a_n=\frac{1}{n^p}$ ($p\in\mathbb{N}$) then the sum is always zero. Also, if $a_n$ grows as fast or faster than $\frac1n$ then the sum converges to zero. Now, one case which did go to $1$ in the limit was
$$a_n=\frac{\log(n^a)}{n}$$
for $a>1$. Unfortunately, I can't tell what happens when $a=1$ (it may very well converge to $1$) but at $a\in\{2,3,1.5,...\}$ it always seems to converge to $1$. 
Again, this is not an answer, but if I were you I would investigate the function $\frac{\log(n^a)}{n}$ and see if that might be some sort of cutoff point.
A: somewhat belated answer, but the necessary and sufficient condition for the coverage is the divergence of the series
$$\sum_{n\geq1}\frac{1}{n^2}\exp\Big(\sum_{l=1}^n a_l\Big).$$
A reference to the result is "Covering the line with random intervals" by L. A. Shepp, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete volume 23, pages 163–170 (1972).
