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  4. The product 2(1/4)⋅ 4(1/16) ⋅ 8(1/48) ⋅ 16(1/128)⋅ ldots to ∞ is equal to :

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Q. The product $2^{\frac{1}{4}}\cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}}\cdot \ldots$ to $\infty$ is equal to :

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Solution:

$2^{\frac{1}{4}}\cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}}\cdot \ldots \, \infty$
$=2^{\frac{1}{4}}\cdot 4^{\frac{2}{16}} \cdot 8^{\frac{3}{48}} \cdot 16^{\frac{4}{128}}\cdot \ldots\,\infty$
$-2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}} \cdot 8^{\frac{1}{16}} \cdot 16^{\frac{1}{32}}\cdot \ldots\,\infty$
$=2^{\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+.......\infty}=\left(2\right)^{\left(\frac{1/4}{1-1/2}\right)}=2^{1/2}$