# Linked Questions

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### If $\sum a_n$ is a convergent series with $S = \lim s_n$, where $s_n$ is the nth partial sum, then $\lim_{n \to \infty} \frac{s_1+…+s_n}{n} = S$ [duplicate]

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 ...
2answers
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### Prove that, if $\lim_{j\rightarrow\infty}a_j=l$ then $\lim_{j\rightarrow\infty}m_j=l$.

So I was wondering if anybody could help me tackle this problem that I've been assigned in my real analysis course (book used is Foundations of Analysis by Steven G. Krantz). From my understanding, ...
1answer
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### Limit of a sequence implication [duplicate]

Be $(a_n)_{n\in N}$ a sequence in $R$ and $a\in R$. Show that $\lim_{n\to \infty}a_n=a ~~~~~\Rightarrow ~~~~~\lim_{n\to \infty} \left(\frac1n \sum_{k=1}^n a_k\right) = a$
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### Question about post http://math.stackexchange.com/questions/1568696/lim-n-rightarrow-infty-a-n-a-how-to-prove-lim-n-rightarrow-infty/1568740#1568740

We understand that Since $(a_n)$ is convergent hence it's bounded. Let $|a_n|\leq K\ \forall\ n\in \mathbb{N}$. Now for $\epsilon >0$ let $N\in \mathbb{N}$ be such that \$|a_n-A|<\epsilon\ ,\...