Consider a case where we have a sequence $x_ka_k$ ($a_k>0$) and $x_k$. We know that $x_k$ converges in $l^p$ and want to know whether $a_kx_k$ converges in $l^q$, or the other way around.
At times, Hölder inequality can be used:
$$\sum_{k=0}^\infty|x_ka_k|\leq (\sum_{k=0}^\infty |x_k|^q)^{\frac{1}{q}}(\sum_{k=0}^\infty |a_k|^q)^{\frac{1}{q}}$$
But I want something more general. It would be quite intuitive, that if $p<q$ and:
$$\sum_{k=0}^{\infty}|x_k|^p$$
converges then: $$\sum_{k=0}^{\infty}|x_k\frac{1}{a_k}|^q$$
converges, where $a_k$ is at least $O(n^{p-q})$. The same should work the other way around, if $q<p$ and:
$$\sum_{k=0}^{\infty}|x_ka_k|^p$$
converges, then:
$$\sum_{k=0}^{\infty}|x_k|^q$$
should converge if $a_k$ is in $O(n^{q-p})$. Is there a proof for such properties or something similar? What theorems to use to prove such facts about specific sequences (the most common sum convergence theorems seem to not work very well)?