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I've been encountering problems that appear simple and similar (as you can see) but I don't have any reference on how to start solving them. My biggest problem, actually, is that a lot of technical terms are different in my language, so I'm having trouble googling for directions. So, even a: "Let me google that for you," would be very helpful!

For the following problems, the function $x \in X$ where $X$ is a Banach space, and $x_n$ is a series of functions.

Problem #1

If we know that: $ \lim\limits_{n\to \infty} x_n = \gamma $

Then, prove the equation:

$$ \lim\limits_{n\to \infty}\frac{x_1+x_3+...+x_{2n+1}}{2n+1} = \frac{\gamma}{2}$$

Problem #2

If we know that: $ \lim\limits_{n\to \infty} x_n = x $

Then, what is:

$$ \lim\limits_{n\to \infty}\frac{x_1+2x_2+...+nx_n}{n^2} $$

Problem #3

If we know that: $ x=(x_n) \in l^1 $ (the $l^p$ space with p-norm, where $p=1$)

Then, prove the equation:

$$ \lim\limits_{n\to \infty}\frac{nx_1+(n-1)x_2+...+2x_{n-1}+x_n}{n} = \sum_{n=1}^\infty x_n $$

avance70
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    I suppose you mean "$n\to\infty$" rather than "$x\to\infty$" everywhere – MPW Jun 11 '15 at 11:59
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    At least some of your questions can be solved using Stolz-Cesaro theorem. You can find several posts which use Stolz-Cesaro theorem on this site. If you have a look at them, that might help you get started. – Martin Sleziak Jun 11 '15 at 12:06
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    I will also add a link to an answer comparing two forms of Stolz-Cesaro theorem. – Martin Sleziak Jun 11 '15 at 12:08
  • I will attempt to use the theorem today. Just a quick question: it says that it's for real numbers; but these problems are all functions over complex numbers... can it still apply? – avance70 Jun 11 '15 at 12:10
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    You should probably mention in your post that you are interested in sequences of complex numbers. It might make a slight difference. Previously I wrote in a comment that Stolz-Cesaro works for complex sequences as well. This is not true, at least not exactly in the form given in the above links. For complex numbers you cannot reasonably talk about monotone sequences and positive numbers. – Martin Sleziak Jun 11 '15 at 12:17
  • Thank you, I added clarifications about the spaces. – avance70 Jun 11 '15 at 12:26
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    However, in your case complex vs. real does not make a real difference. You can only have complex numbers in numerator, in all three problems you have an unbounded increasing sequence in denominator. So you can simply treat real and imaginary part separately. – Martin Sleziak Jun 11 '15 at 12:29
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    @MartinSleziak: Stolz-Cesaro in arbitrary Banach spaces? Does such a thing even exist? – Alex M. Jun 11 '15 at 12:37
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    I haven't heard about it, so I googled and found this vector form of Stolz-Cesaro theorem . – avance70 Jun 11 '15 at 12:41
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    @AlexM Notice that the post has been edited a few times. Originally it did not mention where $x_n$ lives, so I assumed real sequences. Then the OP mentioned complex numbers in comments and edited the post to include Banach spaces. Anyway, since the denominator is in all three cases a real sequence, my guess is that the usual proof of Stolz-Cesaro should work in this case too. Probably, for some of these problems even a weaker result that $a_n\to\ell$ implies $\frac{a_1+\dots+a_n}n\to\ell$ might suffice. – Martin Sleziak Jun 11 '15 at 12:43
  • My mistake, sorry about that :) I though a normed space was implied from it being a functional-analysis question. – avance70 Jun 11 '15 at 12:44
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    @avance70: That form of the Stolz-Cesaro theorem solves all your three problems in a snap. – Alex M. Jun 11 '15 at 12:48

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