Question:

<p><strong>Step 1:</strong> Express each factor as a power of 2:</p><p>$(2)^{1/4} = 2^{1/4}$</p><p>$(4)^{1/8} = (2^2)^{1/8} = 2^{2/8} = 2^{1/4}$</p><p>$(8)^{1/16} = (2^3)^{1/16} = 2^{3/16}$</p><p>Generally, the $n$-th term is: $(2^n)^{1/2^{n+1}} = 2^{n/2^{n+1}}$</p><p><strong>Step 2:</strong> Combine all factors:</p><p>$(2)^{1/4} \cdot (4)^{1/8} \cdot (8)^{1/16} \cdots = 2^{1/4 + 2/8 + 3/16 + 4/32 + \cdots}$</p><p><strong>Step 3:</strong> Evaluate the exponent sum using $S = \sum_{n=1}^{\infty} \frac{n}{2^{n+1}}$:</p><p>Let $T = \sum_{n=1}^{\infty} \frac{n}{2^n} = \frac{1/2}{(1-1/2)^2} = 2$</p><p>Then: $S = \frac{1}{2}\sum_{n=1}^{\infty} \frac{n}{2^n} = \frac{1}{2}(2) = 1$</p><p><strong>Step 4:</strong> Therefore: $(2)^{1/4} \cdot (4)^{1/8} \cdot (8)^{1/16} \cdots = 2^1 = \boxed{2}$</p>