We denote the floor and ceiling functions, respectively, with $\lfloor x \rfloor$ and $\lceil x \rceil$.
We consider the Möbius function, and then the series $$\sum_{n=1}^\infty\frac{\mu\left(\lfloor \sqrt{n} \rfloor\right)-\mu\left(\lceil \sqrt{n} \rceil\right)}{n}.$$
When I've consider this series, the only simple calculation that I can to state is that those terms being a perfect square $m=k^2$ satisfy $$\lfloor \sqrt{m} \rfloor=\lceil \sqrt{m} \rceil=k,$$ and thus for these $m$'s their contribution in the series is $0=\frac{\mu\left(\lfloor \sqrt{m} \rfloor\right)-\mu\left(\lceil \sqrt{m} \rceil\right)}{m}$. But it is a simple observation, and doesn't tell me nothing about the convergence of the series.
I believe that it's an interesting example of series.
Question. Is it possible deduce that this series is convergent $$\sum_{n=1}^\infty\frac{\mu\left(\lfloor \sqrt{n} \rfloor\right)-\mu\left(\lceil \sqrt{n} \rceil\right)}{n}?$$ Many thanks.
Is not required an approximation, only is required the discussion about if the series is convergent or does diverge.
With this my code
sum (mu(floor(sqrt(n)))-mu(ceil(sqrt(n))))/n, from n=1 to 1000
Wolfram Alpha calculate an approximation and show us a graph.