Question

If S = [1+(-1/3)][1+(-1/3)²][1+(-1/3)⁴][1+(-1/3)⁸.....] then S =

Answer 1

Let x = −1/3

S = (1 + x)(1 + x² )(1 + x⁴ )….

(1 − x)S = (1 − x)(1 + x)(1 + x² )(1 + x⁴ )…

(1 − x)S = (1 − x²)(1 + x²)(1 + x⁴)(1 + x⁸ )…..

(1 − x)S = (1 − x⁴)(1 + x⁴ )(1 + x⁸)…..

Since x < 0 & |x| < 1 so the value of RHS would be equals to 1.

(1 − x) S = 1 or S = 1/(1 − x) or 1/(1 −(−1/3)) =3/4.