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 $a_k=0$.

You want to know if $\sum_{k=1}^\infty a_k k^{-n}=0$ for $(a_k)\in l^2$ and every positive integer $n$ implies $a_k=0$. This is true. First we note that if $(a_k)\in l^2$, then $(a_k/k)\in l^1$. So w.l.o.g. we may consider the case where $(a_k)\in l^1$ to begin with. Now 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 $a_k=0$.