+0

# Product

0
36
2

Determine the value of the infinite product

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

May 6, 2022

#1
0

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

May 6, 2022
#2
+32957
+1

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