In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Using a series expansion, show that for all $x > 0$, we have
$$Lf(x) = \frac{1}{2x^2} - \frac{1}{x} + \sum_{n=1}^{+\infty} \frac{1}{(n+x)^2}.$$