Linked Questions

25
votes
3answers
9k views

Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means [duplicate]

Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n}...
2
votes
4answers
329 views

Proving if $\lim_{n\rightarrow\infty}a_n=L $ then $\lim_{n\rightarrow\infty} \frac{a_1+a_2+\cdots+a_n}n=L $ [duplicate]

Good morning, i have a big problem with this proof. I haven't idea about this proof. Problem: Suppose $\lim_{n\rightarrow\infty}a_n =L $ then $\lim_{n\rightarrow\infty}\ \frac{a_1+a_2+\cdots+a_n}...
2
votes
0answers
64 views

Show that if $a_n\rightarrow a$ then $\frac{a_1+…+a_n}{n}\rightarrow a$ [duplicate]

This was an excercise in my exam and I was wondering whether my solution was correct or not. I have said since $a_n\rightarrow a$ there exists a $N$ such that for every $n>N$ we have $|a_n-a|<\...
1
vote
1answer
47 views

Proving that the average approaches zero [duplicate]

If the positive numbers $v_n$ approach zero as $n\to\infty$, prove that their average $(v_1+v_2+\dots+v_n)/n$ also approaches zero. We have: For any $\epsilon$, there is an $N$ such that $v_n<\...
0
votes
0answers
36 views

Prove that, if $\lim_{j\rightarrow\infty}a_j= \ell$, then $\lim_{j\rightarrow\infty}m_j= \ell$. [duplicate]

Let $a_j$ be a sequence of real numbers. Define $$m_j=\frac{a_1+a_2+\dots +a_j}{j}$$. Prove that, if $\lim_{j\rightarrow\infty}a_j= \ell$, then $\lim_{j\rightarrow\infty}m_j= \ell$. Give an example ...
9
votes
3answers
12k views

arithmetic mean of a sequence converges

We had a theorem that the means of a sequence also converges: Let $(a_n)_{n\in\mathbb N}$ be a convergent sequence. Then $\displaystyle \overline a_n=\sum_{k=1}^n \frac{a_k}n$ also converges. I've ...
4
votes
2answers
559 views

If $a_n\to \ell $ then $\hat a_n\to \ell$ [duplicate]

I need some help to finish this proof: THEOREM Let $\{a_n\}$ be such that $\lim a_n=\ell$ and set $$\hat a_n=\frac 1 n \sum_{k=1}^na_k$$ Then $\lim\hat a_n=\ell$ PROOF Let $\epsilon >0 $ be ...
2
votes
5answers
237 views

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 ...
2
votes
2answers
46 views

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, ...
2
votes
1answer
86 views

Limit of a sequence implication [duplicate]

Be $(a_n)_{n\in N}$ a sequence in $ R $ and $a\in R$. Show that $\lim_{n\to \infty}a_n=a ~~~~~\Rightarrow ~~~~~\lim_{n\to \infty} \left(\frac1n \sum_{k=1}^n a_k\right) = a$
0
votes
1answer
42 views

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\ ,\...