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Suppose $a>0$, and we have an infinite series of Bernoulli random variables $B_k$ with $$\mathbb{Pr}{\large[}B_k=1{\large]} = \frac{1}{1+e^{a\cdot 2^k}}$$ Then $$\text{E}\left[\sum_{k=-\infty}^{\...

Let $a_n = \frac{1}{n!}\prod_{i=0}^{n-1} (r+\alpha i)$, for constants $0<r, \alpha<1$. The series is convergent by the ratio test. I want to find the exact value or maybe an upper bound for the ...