Question

2 power 1 by 4 into 4 power 1 by 8 into 8 power 1 by 16 into 16 power 1 by 32 into and so on upto Infinity equals x is simplified then the value of x is

Answer 1

2/4×4/8×8/16×16/32.........x
or 1/2×1/2×1/2×1/2....x
a=1/2 d=0 Tn=?
Tn= a+ (n-1)d
= 1/2+(n-1)0
=1/2

Answer 2

2^1/4 × 4^1/8 × 8^1/16 × 16^1/32 × ...... = x;
qrdrt2 × (2²)^1/8 × (2^3)^1/16 × (2⁴)^1/32 × ..... = x;
sqrt2 × 2^3/16 × 2^1/8 × .... = x;

there is no specific interval so it doesn't belong to any progressive series. hence it is way more difficult to solve.

let, dy = √2 as a small portion of x.
then, x = integration of dy with upper limit √2 and lower limit 0.
x = [y](ul----> √2; ll--->0)
x = √2 [while neglecting smaller values when y<<√2.]