The value of 2^1/44^1/88^1/16 is:

Question

The value of $2^{1/4}\cdot4^{1/8}\cdot8^{1/16}\cdots\infty$ is:

Accepted Answer

The product is $$p = {2^{1/4}}{.2^{2/8}}{.2^{3/16}}........$$

$$ = {2^{1/4 + 2/8 + 3/16 + .......\infty }}$$

Now let

$$S = {1 \over 4} + {2 \over 8} + {3 \over {16}} + .......\infty \,\,\,\,........\left( 1 \right)$$

$${1 \over 2}S = {1 \over 8} + {2 \over {16}} + .......\infty \,\,\,\,........\left( 2 \right)$$

Subtracting $(2)$ from $(1)$

$$ \Rightarrow {1 \over 2}S = {1 \over 4} + {1 \over 8} + {1 \over {16}} + .......\infty $$

or $${1 \over 2}S = {{1/4} \over {1 - 1/2}} = {1 \over 2} \Rightarrow S = 1$$

$$\therefore$$ $P = {2^S} = 2$

Solutions · 1