# Inequality of an Infinite Series

Suppose we have

$$begin{equation} left( sum^infty_{k=0} |a_k|^p right)^{1/p} < + infty end{equation}$$

Is it necessarily true that

$$begin{equation} sum^infty_{k=0} |a_k|^p log|a_k| leq 0 end{equation}$$

Here, for my attempt, I am inclined to say yes. The intuition I have is that under the above condition, we should have that $$|a_k| rightarrow 0$$, which thus implies that $$|a_k|^p log|a_k| rightarrow 0^-$$. If so, how should I go above arguing this rigourously? Thanks!