Let $x_n=\frac{1}{n}$ for $n=1,2,\dots$ Then $x_n$ is not summable but $x_n^r$ is summable for every $r>1$. My question is: Is there a sequence of positive numbers $x_n$ that is summable but $x_n^r$ is not summable for every $0<r<1$?
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