Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Justify the existence of $\int_0^1 \frac{t-1}{\ln t} \,\mathrm{d}t$ and then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n}) \,\mathrm{d}t$.