examday

/TMUA

Let $g_0(x)=x$ for $x\in[0,1)$, and for $n\ge0$ define

(that is, $g_{n+1}(x)$ is the fractional part of $2g_n(x)$). For each $n\ge1$, let

Find the value of $\displaystyle\sum_{n=1}^{100}u_n$.