How to prove it or disprove it? Is it true?
\(\displaystyle \sum_{j=1}^{\infty} \lfloor \frac{n}{2^j }\rfloor=n+1 \; \), \(\displaystyle n,j \in \mathbb{N}\).

Can I ask what exactly you ask them for? They don't seem to be related to a few classes. Are you a tutor getting answers here? I would be fine with that if you were up front about it.

How to prove it or disprove it? Is it true?
\(\displaystyle \sum_{j=1}^{\infty} \lfloor \frac{n}{2^j }\rfloor=n+1 \; \), \(\displaystyle n,j \in \mathbb{N}\).