Prove that:

$\mathrm{LHS}=2^{\frac{1}{4}} \cdot 4^{\frac{2}{8}} \cdot 8^{\frac{3}{16}} \cdot 16^{\frac{4}{32}} \ldots \infty$

$=2^{\left(\frac{1}{4}+\frac{2}{8}+\frac{3}{16} \frac{3}{16} \frac{4}{32} \cdot \infty\right)}$

$=2^{\left(\frac{1}{2^{2}}+\frac{2}{2^{3}}+\frac{3}{2^{4}}+\frac{4}{2^{5}}+\ldots \infty\right)}$

$=2^{\frac{1}{2^{2}}\left\{1+\frac{2}{2}+\frac{3}{2^{2}}+\frac{4}{2^{3}} \ldots \infty\right\}}$

$=2^{\frac{1}{2^{2}}}\left\{\frac{1}{1-\frac{1}{2}}+\frac{1 \cdot \frac{1}{2}}{\left(1-\frac{1}{2}\right)^{2}}\right\}$

$=2^{\frac{1}{2^{2}}\{2+2\}}$

$=2^{1}$

$=2=\mathrm{RHS}$