# Thread: The avarage of {a(n)}

1. ## The avarage of {a(n)}

We know that if real sequence $\displaystyle \left\{a_n\right\}$ converges, then $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges.

Suppose $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges, what one can say about $\displaystyle \left\{a_n\right\}$?

2. Originally Posted by elim
We know that if real sequence $\displaystyle \left\{a_n\right\}$ converges, then $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges.

Suppose $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges, what one can say about $\displaystyle \left\{a_n\right\}$?
Zip.

Let $\displaystyle a_n=(-1)^n$.

Then, $\displaystyle \lim_{n\to\infty}(a_{n+1}-a_n)=\lim\text{ }0$ and clearly $\displaystyle \left|\frac{a_1+\cdots+a_n}{n}\right|\leqslant\fra c{1}{n}\to 0$

3. Originally Posted by elim
We know that if real sequence $\displaystyle \left\{a_n\right\}$ converges, then $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges.

Suppose $\displaystyle \lim_{n \to \infty} (a_{n+1}-a_n) = 0$ and $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converges, what one can say about $\displaystyle \left\{a_n\right\}$?
If $\displaystyle \left\{a_n\right\}$ converges, why does $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converge?

4. Originally Posted by Drexel28
Zip.

Let $\displaystyle a_n=(-1)^n$.

Then, $\displaystyle \lim_{n\to\infty}(a_{n+1}-a_n)=\lim\text{ }0$ and clearly $\displaystyle \left|\frac{a_1+\cdots+a_n}{n}\right|\leqslant\fra c{1}{n}\to 0$
How does $\displaystyle \lim (a_{n+1} - a_{n}) = 0$ when $\displaystyle a_{n+1} - a_{n} = 2$ for n odd and $\displaystyle a_{n+1}-a_{n} = -2$ for n even?

5. Originally Posted by mathman88
If $\displaystyle \left\{a_n\right\}$ converges, why does $\displaystyle \left\{ \frac{a_1+\cdots + a_n}{n}\right\}$ converge?
Actually the latter converges to the same limit.
Given $\displaystyle \epsilon > 0$, there is an $\displaystyle M > 0$ such that $\displaystyle n > M \Rightarrow |a_n-A| < \epsilon$
Let $\displaystyle N = \max \left\{ 2M \max\{|a_1|,\cdots, |a_M|\}+ |A|,M\right\}$, then $\displaystyle n>N \Rightarrow$
$\displaystyle \left| \frac{a_1+\cdots+a_n}{n} -A \right| \le \left | \frac{(a_1-A+\cdots + (a_M - A)}{n} + \frac{(a_{M+1}-A)+ \cdots + (a_n-A)}{n}\right |$
$\displaystyle \le \frac{M(\max\{|a_1|,\cdots, |a_M|\}+ |A|)}{n} + \frac{(n-M)}{n} \frac{\epsilon}{2} < \epsilon$