1. The problem statement, all variables and given/known data I need help in the next questions. Prove or find counterexamples for the next propositions: 1. If the series [ Sigma (from n=1 to infinity) n*an ] converge then the series [ Sigma (from n=1 to infinity) n*a(n+1) ] also converge. 2. If [Sigma (n=1 to infinity) of an ] is a positive converge series then the series [ Sigma (n=1 to infinity) sqrt( an*a(n+1) ) ] converge. 3. IF the series [ Sigma from k=1 to infinity of a(2k-1) ] converge and the series the series [Sigma fron k=1 to infinity of a(2k) ] converge then the seriesl [Sigma fron n=1 to infinity of an] also converge. 4. If lim_n->infinity_ n*an =0 then the series Sigma(an) converge. 2. Relevant equations 3. The attempt at a solution I think that 1 is incorrect but I can't find any counterexample for it. I'm almost sure that 2 and 3 are true, but 4 isn't... Can't find any counterexamples for any of these propositions... Help is NEEDED! TNX everyone!