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Determine the value of the infinite product \((2^\frac{1}{3})(4^\frac{1}{9})(8^\frac{1}{27}) ...\) 

 Aug 9, 2019
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\(\text{It's probably easiest to derive the log of this product and the take the exponential of the log.}\\ p = \prod \limits_{k=1}^\infty \left(2^k\right)^{\frac{1}{3^k}}= \prod \limits_{k=1}^\infty 2^{\frac{k}{3^k}}\\ \log(p) = \sum \limits_{k=1}^\infty \log\left(2^{\frac{k}{3^k}}\right) =\log(2) \sum \limits_{k=1}^\infty\dfrac{k}{3^k} = \\ \log(2) \dfrac{3}{(3-1)^2} = \log(2) \dfrac 3 4\\ e^{\log(p)} = e^{\log(2)\frac 3 4} = 2^{\frac 3 4}\)

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 Aug 9, 2019
edited by Rom  Aug 9, 2019

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