Question

Evaluate $3\times 3^{1/2}\times 3^{1/4}\times 3^{1/8}\times$ ...to $\infty$

Answer 1

$3\times 3^{Y_2}\times 3^{Y_4}\times 3^{Y_8}......\infty$

$=3^{1+Y_2+Y_4+Y_8}$

$\therefore 1+Y_2+Y_4+Y_8$ is one sum of infinite $G.P$

$\therefore Sum=\frac{a}{1-r}$

Where $a=1r=Y_2$

$\therefore 3\times 3^{Y_2}\times 3^{Y_4}........\infty =3^{\frac{1}{1-Y_2}}$

$=3^{Y_{Y_2}}$

$=3^2=9$

$\therefore 3\times 3^{Y_2}\times 3^{Y_4}.....\infty =9$