Compact or noncompact operator?

Question

I found an example of operator: $$A(x_1, x_2, x_3, ...)=\left(x_1+ \frac{x_1+x_2}{4}+\frac{x_1+x_2+x_3}{9}+..., \frac{x_1+x_2}{4}+\frac{x_1+x_2+x_3}{9}+..., \frac{x_1+x_2+x_3}{9}+..., ..., \frac{x_1+x_2+x_3+...+x_k}{k^2}+\frac{x_1+x_2+x_3+...+x_k+x_{k+1}}{(k+1)^2}+...,...\right)$$ I try to understand: is this operator compact or noncompact? I suppose that it is noncompact, but I don't understand how to prove it. I try to use criterion of precompactness in lp(set M is precompactness <=> M is a bounded set and $\forall \varepsilon \exists N: \forall x=(x_1, x_2, x_3,..)\in M $ $ \sum_{k=N}^\infty |(Ax)_k|^p<\varepsilon)$, but I couldn't find a vector x: $||x||=1, \sum_{k=N}^\infty |(Ax)_k|^p>\varepsilon$ to show that this operator is noncompact. Sorry for my English. Thanks!