Find a sequence in $l^p$ but not in $l^q$, where $q < p$

Question

I'm trying to find a sequence that is in $l^p$ but not in $l^q$, where $q < p$. Can anyone help?

Answer 1

I will answer a bit more than what the question actually asks for, since the question itself doesn't really give the full picture.

A function can be in $L^p$ but not $L^q$ for $q<p$ if it has a long tail which larger powers shrink. For instance, for $r<0$, $x^r 1_{[1,\infty)}(x)$ is in $L^p$ if and only if $pr<-1$, i.e. $p>-1/r$.

A function can be in $L^p$ but not $L^q$ for $q>p$ if it has a singularity which larger powers amplify. For instance, if $r<0$ then $x^r 1_{(0,1)}(x)$ is in $L^p$ if and only if $pr>-1$, i.e. $p<-1/r$.

Case 1 is only possible on infinite measure spaces; case 2 is only possible on measure spaces which contain sets of arbitrarily small positive measure. The real line has both of these properties. A bounded interval has just the second property. $\mathbb{N}$ with the counting measure (where the $\ell^p$ spaces live) has just the first property.

Answer 2

What about $x_n=(\frac{1}{n})^{\frac{1}{q}}$?