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Determine the value of the infinite product

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

May 5, 2022

#1
+23242
+1

When multiplying numbers with the same base, add their exponents.

To add  1/3  +  1/9  +  1/27  +  ...

note that it is an infinite geometric sequence whose sum is:  Sum  =  a / (1 - r)

where   a  =  1/3     < the initial term >

and       r  =   1/3    < the common ration >             so:            Sum  =  (1/3) / [ 1 - (1/3) ]

Sum  =  (1/3) / (2/3)

Sum  =  1/2

So:  the product  =  21/2  =  sqrt(2)

May 5, 2022

#1
+23242
+1

When multiplying numbers with the same base, add their exponents.

To add  1/3  +  1/9  +  1/27  +  ...

note that it is an infinite geometric sequence whose sum is:  Sum  =  a / (1 - r)

where   a  =  1/3     < the initial term >

and       r  =   1/3    < the common ration >             so:            Sum  =  (1/3) / [ 1 - (1/3) ]

Sum  =  (1/3) / (2/3)

Sum  =  1/2

So:  the product  =  21/2  =  sqrt(2)

geno3141 May 5, 2022