show that $(x_k)$ is convergent and limit$\notin\ell^p$

Question

Let $\ell^p:=\{(x_n)|\|x_n\|_{\ell^p}<\infty\}$ be the space of sequences with finite $\ell^p$-norm.

Show that the sequence $x_1:=1$, $x_k:=\frac{1}{\log k}$, $k\geq 2$ converges to $x=0$, but $x\notin\ell^p$ for $p\geq 1$.

To show that the sequence converges in $\ell^p$ we have to show that $\|x_n-0\|_{\ell^p}\to 0$. So basically we have to consider the series $\|x_n\|^p_{\ell^p}=\sum_{k=1}^\infty |x_k|^p$. Since a finite number of summands does not change the limit, we can consider the series

$$\sum_{k=2}^\infty \left|\frac{1}{\log k}\right|^p = \sum_{k=1}^\infty \left|\frac{1}{\log(k+1)}\right|^p$$

Now I have to show that the series converges to $0$. For $p>1$ we can use the estimate $\log(k)\leq k-1$ for $k>0$ and hence $$\sum_{k=1}^\infty \left|\frac{1}{\log(k+1)}\right|^p\leq\sum_{k=1}^\infty \left|\frac{1}{k}\right|^p\to 0\quad\mathrm{since\ }p>1$$

But how do I show the convergence for $p=1$? Does the series even converge in this case? And how do I show that $0\notin\ell^p$ (or is this simply because $0$ is a real number and not a sequence?).

Any advice is appreciated, thanks!

Edit: I just realized I made a mistake using the estimate $\log(k)\leq k-1$. From this estimate we get

$$\sum_{k=2}^\infty \left|\frac{1}{\log(k)}\right|^p>\sum_{k=2}^\infty \left|\frac{1}{k-1}\right|^p \quad\begin{cases} \to\infty, & p=1\\ <\infty, & p>1\end{cases}$$

So, having the comments below in mind, for $p=1$ we get that the series does not lie in $\ell^1$. However I still don't know how to show $x_k\notin\ell^p$ for $p>1$.

Answer 1

To show that $(x_n)\not\in \ell^p$, you can use Cauchy-condensation. Then you have: $$\sum_{n=1}^{\infty}\left(\frac{2^n}{\log(2^n)}\right)^p=\sum_{n=1}^{\infty}\left(\frac{2^n}{n\log(2)}\right)^p=\frac{1}{\log(2)^p}\sum_{n=1}^{\infty}\left(\frac{2^{n}}{n}\right)^p=\infty$$

since $\lim_{n\to \infty}\frac{2^n}{n}\neq0.$