The sequence given by: $$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$ That is, $$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$
Is this sequence in the sequence space $\ell_p$ ? where for $1<p< \infty$ and $q>1$:
$$\ell_p= \left \{ \left \{ x_n \right \}_{n\in \mathbb{N}} \subseteq\mathbb{R} \colon \sum_{k=1}^{\infty} \left | x_k \right |^p <\infty\right \}$$