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Determine the value of the infinite product

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

May 14, 2022

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Suppose you mean the exponents form a geometric sequence.

Just combine the exponents using the formula $$a^m a^n = a^{m+n}$$.

$$2^{1/3} \cdot 2^{1/9} \cdot 2^{1/27} \cdots = 2^{1/3 + 1/9 + 1/27 + \cdots}$$

Note that the exponent is now the sum of a geometric sequence. Can you complete it now?

May 14, 2022