= Sinc-squared summation of a sequence tending to zero
{title2=$k\sum_{r\ge1}A_r\operatorname{sinc}^2(rk)\to0$}
If $A_r\to0$, then for nonzero $k$ the weighted series is absolutely convergent and its tail after $n$ is bounded by $2\sup_{r>n}|A_r|/(nk^2)$. This follows from $|\sin(rk)|\le1$ and the reciprocal-square tail bound. For $N=\lfloor1/|k|\rfloor$, the prefix multiplied by $|k|$ tends to zero by splitting at a fixed index where $|A_r|$ becomes small. The multiplied tail is bounded by $2\sup_{r>N}|A_r|/(N|k|)$, which tends to zero. Thus the whole multiplied series tends to zero as $k\to0$. Omitting $k^{-2}$ from the unmultiplied tail bound is invalid.
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