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Determine the value of the infinite product

\((2^{1/3})(2^{1/9})(2^{1/27}) \dotsm\)

  Enter your answer in the form "\sqrt[a]{b}"

 May 6, 2022
 #1
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productfor(n, 1, infinity, (2^(1/(3^n)))==It converges to sqrt(2)

 May 6, 2022
 #2
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Note that \(2^a2^b2^c....=2^{a+b+c...}\)

 

In your case \(a+b+c\) forms an infinite geometric series, ie. \(b = ar, c = ar^2, ...etc\)

 

The limit of the infinite geometric series here is \(\frac{a}{1-r}\) or \(\frac{1/3}{1-1/3}\) 

 

You can complete the calculation.

 May 7, 2022

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