Series — bounding a grouped harmonic sum

The value of $1+\left(\frac{1}{2^1}+\frac13\right)+\left(\frac{1}{2^2}+\frac15+\frac16+\frac17\right)+\cdots+\left(\frac{1}{2^9}+\cdots+\frac{1}{1023}\right)$ lies between