Find a sequence converging to zero but not the element of $\ell^p$ space for any $1I am studying functional analysis and I have a problem about finding a sequence converging to zero such that this sequence is not in $\ell^p$ for any $p$. By $\ell^p$, I mean
$$\ell^p := \left\{ (x_k)=(x_1,x_2,...):\sum_{k=1}^\infty|x_k|^p<\infty \right\}$$ where $1<p<\infty$.
First, I thought of the simple sequence $(1/k)_{k \in \mathbb{N}}$ which converges to zero, but, then, I realized it is an element of $\ell^p$ when $p>2$. I thought of a couple more examples, but they did not work either. Can somebody help me out here?

Try the sequence $x_k = 1/\ln(k+1)$.
