Question

The value of $ 2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}...\infty $ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

$2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \ldots \infty$
$=2^{\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\ldots \infty}$
$=2^{\frac{1}{2^{2}}\left(1+\frac{2}{2}+\frac{3}{2^{2}}+\frac{4}{2^{3}}+\ldots\right)}$
$=2^{\frac{1}{2^{2}}\left(\frac{1}{1-\frac{1}{2}}-\frac{1 \cdot \frac{1}{2}}{\left(1-\frac{1}{2}\right)^{2}}\right)}$
$=2^{\frac{1}{2^{2}}|2+2|}=2^{1}=2$