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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!

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    $\begingroup$ $x_n = \frac{1}{n \log n}$ (or shift it left if you insist on the sequence starting at $n=0$ or at $n=1$). $\endgroup$ – Daniel Schepler Jun 20 '17 at 0:13
  • $\begingroup$ Wow! That just solved it! Don't you want to post this as an answer so I can accept it? $\endgroup$ – Gabriel Jun 20 '17 at 0:17
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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.)

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