Apply Stolz-Cesaro theorem: Stolz-CesÃ*ro theorem - Wikipedia, the free encyclopedia
Let .
is a sequence with positive terms, strictly ascending and unbounded.
Then .
So,
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?
Apply Stolz-Cesaro theorem: Stolz-CesÃ*ro theorem - Wikipedia, the free encyclopedia
Let .
is a sequence with positive terms, strictly ascending and unbounded.
Then .
So,
Well,
so this can't work.
This argument may not quite work either. Someone will have to double check me on this.
As you suggested, write this as
Now, because we know that for some n = N (where N is sufficiently large) that is arbitrarily close to A. Similarly for a sufficiently large M (not necessarily the same as N, but we may take it to be as least as large as N) the fraction
is arbitrarily close to . Thus:
Now pick the larger of N and M and call it P. Then
I think this will work, but I may have skipped crucial details.
-Dan