JEE Probability

If $A_{1},A_{2},A_{3}.....A_{1006}$ be independent events such that $P(A_{i})=\frac{1}{2i}(i=1,2,3....1006)$ and probability that none of the events occurs be $\frac{\alpha!}{2^{\alpha}(\beta!)^{2}}$ , then

$A)$ - $\beta$ is of form $4k+2$ where $k$ is a positive integer.

$B)$ - $\alpha=2\beta$

$C)$ - $\beta$ is a composite number

$D)$ - $\alpha$ is of form $4k$ where where $k$ is a positive integer.

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