Inequality over infinite sum I have been dealing with this inequality: For some positive real numbers $x_1,...,x_n$ such that $\prod^n_{j=1}x_j>1$ Prove that: $$\sum_{k=1}^{\infty}\frac{\sum^{n}_{j=1}x_j^{k+1}}{\sum_{j=1}^{n}{x^{2k+1}_j}}\leq \frac{1}{\sqrt[n]{\prod^n_{j=1}{x_j}}-1}$$ Since there is a infinite sum, I though about trying some kind of induction over the sum first sum ($\sum^{m}_{k=1}$) and chack if it is correct for every $m$, it as to be for $\infty$. The thing is that I can't solve it. Any solution would be appreciated.

The point is that if $x>1$, then $$ \frac{1}{x-1}=\sum_{k=1}^\infty x^{-k}. $$ Therefore it is sufficient to show that $$ \frac{\sum^{n}_{j=1}x_j^{k+1}}{\sum_{j=1}^{n}{x^{2k+1}_j}}\leqslant \frac{1}{\sqrt[n]{\prod^n_{j=1}{x_j^k}}}. $$ This is not hard now.