Show that $$\prod_{i=1}^\infty(1-\frac1{2^i})>0.288$$
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1http://math.stackexchange.com/questions/78689/the-limit-of-infinite-product/ – Sep 29 '13 at 14:33
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1This product also appeared here. – Marko Riedel Sep 29 '13 at 21:40
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To get the required bound, you need to modify my argument from the other answer: $$\prod_{k=1}^\infty \left({1-{1\over 2^k}}\right) \geq \prod_{k=1}^5\left({1-{1\over 2^k}}\right) \left(1-\sum_{k=6}^\infty{1\over 2^k}\right) ={315\over 1024}\cdot {15\over 16}={4725\over 16384}\approx .28839.$$