# Convergence of sequences

1. Sep 11, 2011

### l888l888l888

1. The problem statement, all variables and given/known data
let (An) be a sequence in R with |summation from n=1 to infinity(An)|< infinity. Prove lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0

2. Relevant equations

3. The attempt at a solution
I think |summation from n=1 to infinity(An)|< infinity means the summation converges .I rewrote " lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0 " as "lim as n goes to infinity of ((summation from k=1 to n of nAn)/n)=0. Since I assmumed "summation from n=1 to infinity(An)" converges, that would imply the lim of An is 0. I dont know how to use this info to prove what i need to prove. Hope you can understand this. I dont know how to use tek.

2. Sep 11, 2011

### noc_njz

Hint: If the summation of An converges and for all n ,Bn<An, then the summation of Bn converges.

3. Sep 11, 2011

### noc_njz

Forgive my missing absolute. |Bn|<|An|