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Let $R>2$ be a real number. Then, for any $n\geq 1$, it holds that

$$ \frac{ \exp(n R)+ 1}{\exp( n R) - \exp(\pi R/2)} \leq \exp( R/n^2).$$

How do I prove this?

  • 2
    For $n=1$ the inequality: $$e^R+1\leq e^R(e^R-e^{\pi R/2})$$ is false, because $e^{(1+\pi/2)R}\geq e^{2R}-e^R-1$ for $R\geq 0$.2011-12-05

1 Answers 1