We prove that $\sum _{k=1}^{\infty }\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)\frac{k^{k-2}}{{(x+k)}^{n+k}}<\frac{1}{x^{n+1}}$holds for all integers $n\ge 0$ and real numbers $x>0$. This extends a result of Ramanujan, who submitted the inequality with $n=0$ as a problem to the “Journal of the Indian Mathematical Society”.

中文翻译：

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拉马努金不等式的推广

我们证明$\underset{k=1}{\overset{\mathrm{无穷大}}{\Sigma }}\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)\frac{k^{k-2}}{{（X+k）}^{n+k}}<\frac{1}{X^{n+1}}$对所有整数都成立$n\ge 0$和实数$X>0$。这扩展了拉马努金的结果，他提出了不等式：$n=0$作为“印度数学会杂志”的一个问题。