Sum is less than 1.333..

This is trivial ...

consider the terms in the expansion from

1/(2^k)² .... 1/(2^k+1-1)²

now each term in the series is less than equal to the first term ...

let Sk be their sum

Sk<=1/(2^k)² + 1/(2^k)²+ ..... a total of 2^k+1-2^k terms ..

so

Sk <=(2^k+1-2^k)/(2^k)² <= 1/2^k (cancelling 2^k from numerator and denominator)

now Sn = S1 + S2 + S3+ .... Sk + .. some remainder term ....
≤S1+S2+.....+S(k+1)
≤1/2 + 1/4 + 1/8 + .... + 1/2^k+1
≤1/2 + 1/4 ... upto infinity
≤1/2/(1-1/2)
≤1