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If $x$ is sequence such that $x_k=\sin(\frac{1}{\sqrt[3]k})-\sin(\frac{1}{\sqrt[3](k+1)})$ are coordinats, prove that $x \in l_p$ for every $p \ge1$

So, I have to prove $\sum_k (\sin(\frac{1}{\sqrt[3]k})-\sin(\frac{1}{\sqrt[3](k+1)}))^p$ is convergent. I have approximation for p>2, but need help with $1 \le p \le 2$

Mittens
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stranger
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For $p=1$ the series is trivially convergent because it telescopes. $p>1$ is dominated by $p=1.$ (the key observation is that the terms are eventually positive, so we can drop the absolute values).

Igor Rivin
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