TITLE: What can we say about $\sum x_n$ if $\lim n x_n =0$?
QUESTION [5 upvotes]: I proved that if $(x_n)$ is a decreasing sequence of non-negative real numbers such that $\sum x_n$ converges, then $\lim nx_n=0$.
I ask about the other direction. If $(x_n)$ is any sequence of real numbers such that $\lim nx_n=0$ it is true that $\sum x_n$ converges?
What if $(x_n)$ is also a decreasing sequence of non-negative real numbers?
It seems to me that the answer to both of these questions is no! So I searched counter-examples. Unfortunatelly I couldn't find any. Neither could I prove these statements. Can someone help me? Thanks!

REPLY [6 votes]: No to both: $x_n = \frac{1}{n \log n}$ is a counterexample to both.  (To get around the fact $x_1$ is not defined, you can make a definition by cases to set e.g. $x_1 = 10$ and $x_n = \frac{1}{n \log n}$ otherwise.)