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}"
+23252 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)
+23252 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)
