jasperE3 wrote: Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$. Note that $b_k=2\left\{\frac {\left\lfloor 2^k\frac{\sqrt 2}2\right\rfloor}2\right\}$ is the $k^{th}$ digit right of binary dot in the binary representation of $\frac{\sqrt 2}2$ So $a_n=\sum_{k=1}^nb_k2^{-k}$ is $\frac{\sqrt 2}2$ truncated after $n$ digits (right of binary dot). Hence convergence and $\boxed{\lim_{n\to+\infty}a_n=\frac{\sqrt 2}2}$