Here goes: I know that the limit as n goes to infinity of (

an) = A. I need to find the limit of the following as n goes to infinity:

(a1 + 2a2 + 3a3 + · · · + nan)

---------------------------------------

[n(n+1)] / 2

OK. I know the limit goes to A. And I know that (1 + 2 + 3 + ... + n) = [n(n+1)] / 2. So I want to say that this limit equals

(1 + 2 + 3 + ... + n)*(an)

------------------------, which would be just (an) and therefore its limit is A

[n(n+1)] / 2

Although I'm not sure if this works, since I would be multiplying each coefficient by an and not by a1, a2, or whatever.

Is there a different way to show this?