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With $a_n = 1/(n \log n)$ and $b_n(x)=\sin(2\pi n x)$, we have $a_n$ decreasing monotonically and converging to $0$ uniformly with respect to $x \in [\delta,1]$ with $\delta > 0$. Also $b_n(x)$ has uniformly bounded partial sums. If $2x \neq m\in \mathbf{N},$ $$\left|\sum_{k=2}^n \sin(2\pi k x)\right|\leq 1+ \left|\frac{\cos(\pi x)-\cos[2(n+1/2)\pi ...



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