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.
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...
Well...I am pretty bad at counterexample but:
in 1- I think that a series that include non-positive elements might do the work but I can't figure out how to construct it... Each try gives me a series that converges also for n*a(n+1)...
In 2-I am pretty sure this proposition is correct but I can't find an elegant way to prove it...
IN 3- It's pretty obvious that Sigma_an = Sigma_a(2k) +Sigma_a(2k-1) ...But is it realy the proper proof? Is it a right way?
In 4- I'm pretty sure it's incorrect but again, each try gives me an incorrect counterexample...