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For $$k\geq 8$$ show that $$\displaystyle {\sum_{n=1}^{\infty} \left( \left[\frac{2^{k+1}}{3^n}\right]-2\left[\frac{2^k}{3^n} \right] \right) > 0}\quad$$ where $$[x]$$ is the floor function $$\forall x \in \mathbf{R} \left([x] \in \mathbf{Z}\cap (x-1,x]\right)$$