# An example of a sequence in $l_4$ but not in $l_3$?

Can you provide an example of a sequence in $l_4$ but not in $l_3$?

• Hint: Try to think about the harmonic series.
– user99914
Nov 20 '15 at 13:01
• What about $\frac{1}{n^{1/3}}$? Nov 20 '15 at 13:03
• That is an extremely good example. Nov 20 '15 at 14:02

Hint. We have $x := (\frac 1n) \in \bigcap_{p > 1}\ell^p \setminus \ell^1$. Now consider $(x_n^q)$ for the right $q$.