$\displaystyle for \ n \in N \ , \ let \ x_n=1+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+.. .+\frac{1}{n^3} \ and \ let \ n_k=2^{k}-1 \ , k \in N $
$\displaystyle prove \ that \ x_{n_k} < 1+\frac{1}{4}+(\frac{1}{4})^2+(\frac{1}{4})^3+.... . +(\frac{1}{4})^{k-1} , \forall k=2,3.4,... $