a series of general term (un) is convergent but n*(un) doesn't go to zero as n goes to infinity

Question

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!