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# help

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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$$\sqrt[a]{b}.$$

Feb 16, 2020

#1
-1

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
+1

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

Feb 16, 2020
#3
+29257
+5

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