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!