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How do I evaluate $\displaystyle\lim_{k\to\infty} \prod_{i=1}^{k}(1-\alpha_i+\alpha_i^2)$?

Here, $\alpha_k\in (0,1)$ for every $k\in\mathbb{N}$ and $\displaystyle\lim_{k\to\infty}\alpha_k=0$.

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    Deat Robert. Can you give me a proof of your statement?2012-03-17

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The limit of the products $\prod\limits_{i\leqslant k}(1-\alpha_i+\alpha_i^2)$ when $k\to\infty$ is a nonnegative number in $[0,1)$, which is positive if and only if the sum of the series $\sum\limits_k\alpha_k$ is finite.

Edit: The WP page on infinite products might prove useful and, to begin with, the first section on convergence criteria.

Once this is ingested, one can come back to the question here. Consider $\beta_k=\alpha_k-\alpha_k^2$. For every $\alpha_k$ in $(0,\frac12)$, $\beta_k\leqslant\alpha_k\leqslant2\beta_k$. Since $\alpha_k\to0$ by hypothesis, this proves that the series $\sum\limits_k\alpha_k$ converges if and only if the series $\sum\limits_k\beta_k$ does.

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    See Edit. (But to call any of this *my fact* is inappropriate.)2012-03-17