Suppose a series as .
(That is to say has a limit point at infinity.)
Prove that
I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series I am having an embarrassing time showing the result at all.
Another question: Does this theorem have an actual use? I can't think of one.
(FYI: This problem is actually designed for the series to be complex. I figure if I can find out how to prove it for real series then I can generalize it to complex numbers myself.)
Thanks!
-Dan
Thank you for the effort, but this statement is incorrect. Perhaps the mistake in terminology Plato pointed out is a problem. So let me restate the original problem in more correct terms:
Suppose a sequence has the property as .
(That is to say has a limit point at infinity.)
Prove that
Since we are talking about a sequence here we cannot say that for all n, we are saying that the sequence approaches a limit point , as n approaches infinity. That is to say there is only one sequence, not an infinite number of sequences tending toward the same limit.
-Dan
I am somewhat confused almost from the beginning by the line
Is this again a case of my mistaken wording? If you define to be the nth partial sum of a series, this statement is quite understandable and falls out naturally. (Hopefully by now it is clear that this is not what I meant.)
-Dan
There's a neat proof of this by CaptainBlack here for the case where α=0. You can easily deduce the general case from this by applying it to the sequence .
This is a case of Cesàro summation, which is an essential tool in Fourier theory. The main result about convergence of Fourier series (Fejér's theorem) is proved by using Cesàro summation.
So yes, the fact that ordinary convergence implies Cesàro convergence is certainly useful.