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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}"

Guest May 6, 2022

Guest · Answer 1 · 2022-05-06T06:55:20

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

Alan · Answer 2 · 2022-05-07T07:39:27

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.