i have been struggling to find an example of a positive convergent series of general term (un), such that n*(un) doest go to 0 as n goes to infinity, i was able to prove that if (un) is a decreasing subsequence, then n*(un) goes to zero when n goes to infinity, but i can't think of a sequence that isn't monotonous and checks the conditions above. Any help is appreciated. Thanks!
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For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Martin R Feb 05 '23 at 13:58
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You can modify the example here. – David Mitra Feb 05 '23 at 13:58
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Do you really mean a series $\Sum_{i=1}^{\infty}u_n$ or a sequence $u_n$ ? – Jean Marie Feb 05 '23 at 15:42