Searching for a particular kind of convergent sequence

[A-ManESL]

I want an example of a complex sequence $$(x_n)$$ which converges to 0 but is not in ℓ$$^p$$, for $$p\ge 1$$ i.e. the series $$\sum |x_n|^p$$ is never convergent for any $$p\ge 1$$. Can someone provide an example please?

 

[snipez90]

I suspect 1/log(n+1) will work, but I haven't checked divergence for p > 1.

*EDIT* I'm fairly certain it works. I'll let you figure out the estimates needed to demonstrate divergence (hint: you don't need obscure series tests).

 

[A-ManESL]

I am having trouble establishing the divergence. Can you be more explicit? Thanks.

 

[mathman]

For divergence I would guess (1/log(n))^p > 1/n for sufficiently large n.