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Determine the value of the infinite product \((2^{1/3})(4^{1/9})(8^{1/27})(16^{1/81}) \dotsm.\)Enter your answer in the form "\sqrt[a]{b}", which stands for\(\sqrt[a]{b}.\)

 

 

Thanks for your time! :)

 Feb 16, 2020
 #1
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By Mathematica, the product is Prod[(2^n)^{1/3^n}, n = 1, n = n + 1, n <= inf], which spits out \(\sqrt[3]{4}\), or \sqrt[3]{4}.

 Feb 16, 2020
 #2
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∏ [ (2^n)^(3^-n), n, 1, ∞ ] =2^(3/4) =(2^3)^(1/4)

 Feb 16, 2020
 #3
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My attempt is as follows:

 



 

Edit: common ratio should be lambda^-1; the RHS of the second equation on the line beginning "hence" should have a 1 in the numerator not a lambda.

 Feb 16, 2020
edited by Alan  Feb 16, 2020

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