Is there a sequence in $\ell^1$ but not in $\ell^p$ for $p<1$?

Question

I was wondering: is there a sequence of positive numbers $(a_n)$ such that $\sum a_n<\infty$ but for every $0<p<1$, $\sum a_n^p=\infty$?

None of the convergent series I was able to come up with fulfilled this property. But it seems to me that there must be one. Any suggestions?

Answer 1

Take, for example, $$ a_n=\frac{1}{n\log^2 n} $$ for $n\ge2$.