Lucas Numbers Inequality

Question

Can it be shown that

\begin{align} \frac{1}{\ln(1+L_{n}) -1} \geq \frac{L_{n}}{(L_{n}-1)(e^{L_{n}}-1)} \end{align}

where $L_{n}$ is the $n^{th}$ Lucas number. Show results in full detail.

Answer 1

$ln(1+L_n) \leq L_n \to ln(1 + L_n) - 1 \leq L_n - 1 \to \dfrac{1}{ln(1 + L_n) - 1} \geq \dfrac{1}{L_n - 1}$.

Also: $e^{L_n} \geq L_n + 1 \to e^{L_n} - 1 \geq L_n \to 1 \geq \dfrac{L_n}{e^{L_n} - 1}$.

The conclusion follows from these inequalities.