TITLE: a sequence $\{s_n\}$ with $\sum s_n$ convergent
QUESTION [10 upvotes]: what would be $\{s_n\ge0\}$  such that 
$$\sum_{n=1}^\infty s_n$$ converges
but 
$$\lim_{n\to\infty}(n s_n) \neq 0$$

REPLY [3 votes]: Answers are up, this is more a meta-answer.
The problem is false as presented.  The notation "$\displaystyle \lim_{n \to \infty} na_n \neq 0$" means that the limit exists and is nonzero, and if (as stated in the question) the terms of the sequence are positive, that implies $\sum a_n$ is infinite.   The correct statement is "$na_n$ does not converge to zero", which includes the case where no limit exists.  
The similarity between the posted answers is explained by the fact that in any solution there exists an infinite subsequence where $na_n$ is at least equal to a constant but for which $\sum a_n$ restricted to the subsequence is finite.  For construction of examples, it is necessary and sufficient to specify such a subseries, and fill the rest of the terms with any convergent series.