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You want to know if $\sum_{k=1}^\infty a_k k^{-n}$ for $(a_k)\in l^2$ and every positive integer $n$ implies $a_k=0$. This is true. First of all, note that the condition implies that $\sum_{k=1}^\infty a_kp(1/k)=0$ for every polynomial $p$ with $p(0)=0$. Now use the Weierstrass approximation theorem to conclude that $\sum_{k=1}^\infty a_k f(1/k)=0$ for every continuous function $f$ on $[0,1]$ with $f$ vanishing near the origin. It follows easily that $a_k=0$.