Suppose the positive sequence $p_1,p_2,p_3,\ldots$ satisfies $\displaystyle\sum\limits_{i=1}^\infty p_i=1$. Prove that if $\displaystyle\sum\limits_{i=1}^\infty p_i \log i$ converges, then so does $\displaystyle\sum\limits_{i=1}^\infty p_i \log \frac1{p_i}$.
I found this problem in an information theory textbook (problem 2.45 from Elements of Information Theory by Cover and Thomas), but the problem can be formulated as a pure math problem. If you're curious, the original problem is: Show that for a discrete random variable $X\in \{1,2,\ldots\}$, if $\mathbb E\left[\log X\right]<\infty $, then $H(X)<\infty$. ($H(X)$ is the entropy of random variable $X$.)