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Let $\sum a_n$ be a convergent series, and let $S = \lim s_n$, where $s_n$ is the nth partial sum.

I need to prove the following:

$\lim_{n \to \infty} \frac{s_1+...+s_n}{n} = S$

How do I go about proving that proof?

Definition of a limit

$\lim_{n \to \infty} f(x) = L$ if for every number $\epsilon>0$ there is some number $\sigma >0$ such that $|f(x)-L| <\epsilon$ whenever $0<|x-a|<\sigma$

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    What are your thoughts on this? Start with the definition of limit. – 2017-01-18
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    So as n approaches infinity I need to show that it is equal to S – 2017-01-18
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    Of course. I suggest you put up the definition of limit that you know, in your question, so that we are on the same wavelength. Once you know the definition of limit, you will know the starting point. I am trying to take you slowly through this, so that you understand and can solve similar questions later in this way. – 2017-01-19
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    just edited my definition of a limit into the question – 2017-01-19
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    Observe the below answer. It starts with taking $\epsilon>0$, and then uses the fact that $s_n \to S$ to extract an $N$. Some playing around then gives the answer. Don't look at the playing around, just the use of the limit definition should be observed in the below answer. That is your lesson: first, proceed from definition. – 2017-01-19
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    @астонвіллаолофмэллбэрг I agree, and would also observe that intuition plays a huge role. Think about why it's true. The limit in question is just a running average. Since $s_n$ eventually settles to being very close to $S,$ the infinite number of partial sums $s_n$ very close to $S$ in tail will make the average come out to $S$, overwhelming whatever finite stuff happens at the beginning. The proof is just making this idea precise. (But as you say, in figuring out how to make it precise, you need to start from the precise definitions of 'what you know') – 2017-01-19
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    @spaceisdarkgreen Thank you for your comment. I have upvoted your answer, because it makes clear the idea you have written in your comment. – 2017-01-19
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    @астонвіллаолофмэллбэрг Sure, thanks! The comment was as much for OP as for you. – 2017-01-19
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    There already are several posts on this site about this: http://math.stackexchange.com/questions/1439987/suppose-lim-limits-n-to-%E2%88%9E-a-n-l-prove-that-lim-limits-n-to-%E2%88%9E-frac http://math.stackexchange.com/questions/1568696/how-to-prove-that-if-lim-n-rightarrow-inftya-n-a-then-lim-n-rightarr http://math.stackexchange.com/questions/207910/prove-convergence-of-the-sequence-z-1z-2-cdots-z-n-n-of-cesaro-means http://math.stackexchange.com/questions/248116/arithmetic-mean-of-a-sequence-converges http://math.stackexchange.com/questions/930436 – 2017-01-19

5 Answers 5

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Let $\epsilon>0.$ Since $s_n\rightarrow S$ we can choose an $N$ such that $s_n>S-\epsilon$ for all $n>N.$ Let $n>N$. Then can write $$ \frac{s_1 + \ldots + s_n}{n} = \frac{s_1 + \ldots + s_N}{n} +\frac{s_{N+1}+\ldots s_n}{n} > \frac{s_1 + \ldots + s_N}{n} + \frac{n-N}{n}(S-\epsilon).$$ Taking $\liminf$ of both sides gives $$\liminf_{n\rightarrow\infty}\frac{s_1 + \ldots + s_n}{n} \ge (S-\epsilon).$$ This is true for all $\epsilon>0,$ so we have $$\liminf_{n\rightarrow\infty}\frac{s_1 + \ldots + s_n}{n} \ge S.$$

By the same reasoning, can show that $$\limsup_{n\rightarrow\infty}\frac{s_1 + \ldots + s_n}{n} \le (S+\epsilon)$$ for any $\epsilon >0.$ so that $$\limsup_{n\rightarrow\infty}\frac{s_1 + \ldots + s_n}{n} \le S.$$ Since $\liminf \le \limsup,$ these inequalities yield $\liminf = \limsup = S,$ so the limit exists and it is equal to $S$.

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    so does this show that the limit is S? – 2017-01-19
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    @jeb650 I think so. What do you think? – 2017-01-19
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    just making my way through your answer and trying to understand it – 2017-01-19
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    @jeb650 If it helps, remember we're fixing $N.$ The fact that it's $\ge S-\epsilon$ for any $\epsilon >0$ means that it's $\ge S.$ – 2017-01-19
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    I think that last edit makes thinks more clear, thanks, that looks like a correct solution to my question – 2017-01-19
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    @jeb650 Yeah, sorry, it was a little weird cause of the epsilons. Depending how rigorous you want to be, there might be a slight issue with assuming the limit exists but you can replace with liminf (for the first one) and limsup (for the the second) with no issues and the same conclusion. – 2017-01-19
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    so is there a error in the solution? – 2017-01-19
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    I think you should take $\lim\sup$ and $\lim\inf $ instead of limit and then take the difference in the inequality which will give you that $\lim\sup\frac{s_1 + s_2 + ..... + s_n}{n} = \lim\inf \frac{s_1 + s_2 + ..... + s_n}{n} $ hence the limit exists. Then the full proof will follow. – 2017-01-19
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    @jeb650 since it appears that we care, I will fix. – 2017-01-19
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    thank you very much for your time – 2017-01-19
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Using the Stolz-Cesaro Theorem, we have

$$\lim_{n\to \infty}\frac{\sum_{k=1}^n\left(\sum_{j=1}^ka_j\right)}{n}=\lim_{n\to \infty}\sum_{j=1}^{n+1}a_j=S$$

If one is unfamiliar with Stolz-Cesaro, then on can proceed as follows. Let $\epsilon>0$ be given. Then, choose and fix $N$ so large that $|s_k-S|<\epsilon/2$ whenever $k>N$..

Thus, we can write

$$\begin{align} \left|\frac{\sum_{k=1}^n s_k}{n}-S\right|&=\left|\frac1n \sum_{k=1}^n (s_k-S)\right|\\\\ &\le \frac1n \sum_{k=1}^N |s_k-S|+\frac1n \sum_{k=N+1}^n |s_k-S|\\\\ &\le \frac1n \sum_{k=1}^N |s_k-S| +\left(1-\frac{N}{n}\right)\frac{\epsilon}{2}\\\\ &\le \frac1n \sum_{k=1}^N |s_k-S| +\frac{\epsilon}{2}\\\\ &\le \frac{\epsilon}{2}+\frac{\epsilon}{2}\\\\ &=\epsilon \end{align}$$

whenever $n>\max\left(N+1,\frac{2\sum_{k=1}^N |s_k-S|}{\epsilon}\right)$

And we are done!

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    thanks for your answer, but we have not touched on this theorem before. Is there another explanation? – 2017-01-18
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    Well, we can write $\sum_{k=1}^n s_k=\sum_{k=1}^N s_k+\sum_{k=N+1}^n s_k$. Then, given $\epsilon>0$, we choose and fix $N$ so that $S-\epsilon – 2017-01-19
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    I've added another way forward. Please let me know how I can improve my answer. I really want to give you the best answer I can. -Mark – 2017-01-22
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The point is that:

$$\big\lvert\frac{s_1+s_2+\ldots+s_n}{n}-S\big\rvert=\big\lvert\frac{s_1-S+s_2-S+\ldots+s_n-S}{n}\big\rvert\le\frac{1}{n}(\lvert s_1-S\rvert+\lvert s_2\rvert-S+\ldots+\lvert s_n-S\rvert)=\frac{1}{n}\sum_{i=1}^N\lvert s_i-S\rvert+\frac{1}{n}\sum_{i=N+1}^n\lvert s_i-S\rvert$$

The first term goes to zero when $n\rightarrow\infty$

By choosing $N$ large enough one can bound each of $\frac{1}{n}\sum_{i=1}^N\lvert s_i-S\rvert$ and $\frac{1}{n}\sum_{i=N+1}^n\lvert s_i-S\rvert$ by $\frac{\epsilon}{2}$

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$$\left|\frac{1}{n}\sum_{k=1}^n s_k - \sum_{k=1}^n a_k\right| =\left|\sum_{k=1}^n (\frac{n-k+1}{n} a_k - a_k) \right|$$ $$ \leq \varepsilon \left| \sum_{k=1}^m a_k\right| + \left| \sum_{k=m+1}^n a_k\right|\leq \varepsilon (|S|+\delta) + \varepsilon'$$ where for any given $\varepsilon,\varepsilon',\delta >0\ $, $m$ can be chosen so that the above holds for all sufficiently large $n$. Namely: First choose $m$ large enough so that $\left|\sum_{k=1}^m a_k - S\right| < \delta$ and $\left| \sum_{k=m+1}^n a_k\right|<\varepsilon'$. Next, choose $n_0$ large enough so that for all $n\geq n_0$ it is true that $ |\frac{1-k}{n}| < \varepsilon$ for $k\leq m$. Thus, the difference of the two series becomes arbitrarily small; since one of them converges to $S$, the other one also converges to $S$.

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You re-prove Cesaro's theorem. By definition there exists $N\in \mathbf N$ such that $$\lvert S_n-S\rvert <\varepsilon\quad\forall n>N. $$ We deduce that, for $n>N$, \begin{align} \biggl\lvert\frac{S_1+\dots+S_n}n-S\,\biggr\rvert&=\biggl\lvert\frac{S_1-S+\dots+S_N-S}n+\frac{S_{N+1}-S+S_n-S}n\,\biggr\rvert\\ &\le\biggl\lvert\frac{S_1-S+\dots+S_N-S}n\biggr\rvert+\biggl\lvert \frac{S_{N+1}-S}n\biggr\rvert+\dots+\biggl\lvert \frac{S_n-S}n\,\biggr\rvert\\ &\le\biggl\lvert\frac{S_1+\dots+S_N}n-\frac{NS}n\biggr\rvert+\biggl\lvert \frac{S_{N+1}-S}n\biggr\rvert+\dots+\biggl\lvert \frac{S_n-S}n\,\biggr\rvert\\ &\le\frac{\lvert S_1+\dots+S_N+NS\rvert}n+\frac{\lvert S_{N+1}-S\rvert}n+\dots+\frac{\lvert S_n-S\rvert}n\\ &\le\frac{\lvert S_1+\dots+S_N+NS\rvert}n+\frac{\varepsilon}n+\dots+\frac{\varepsilon}n =\frac{\lvert S_1+\dots+S_N+NS\rvert}n+\frac{(N-n)\varepsilon}n\\ &\le \frac{\lvert S_1+\dots+S_N+NS\rvert}n+\varepsilon\le 2\varepsilon \end{align} if $n$ is large enough.