Infinite product calculation

Question

I want to calculate

$ \displaystyle \prod_{n=1}^{\infty} (1- {{c_i}\over a}) $ where $\displaystyle lim_{i\rightarrow \infty} c_i=0$ and $a$ is a small positive constant

Is the information I provided enough? It must converge to zero.

I can't seem to find the euler products of any use. I also tried to prove that the power series$ \displaystyle \sum_{n=1}^{\infty} \log(1-{{c_i}\over a}) $ converges to $-\infty$.

Answer 1

By Taylor,

$$\log(1-c_i)\approx-c_i-\frac{c_i^2}2-\cdots.$$

So if $c_i=\Omega\left(\frac1n\right)$, the series goes to $-\infty$.

Answer 2

Hint. Notice that since $e^x \geq 1 + x$ for any real $x$, you can bound $\prod_{i=1}^\infty (1 - \frac{c_i}{a}) \le \prod_{i=1}^\infty e^{-\frac{c_i}{a}} = \exp[-1/a \sum_{i\geq 1} c_i]$.