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I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let $$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & 1 & 0 & \...

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...

The series $\sum_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)...