Converging Sequence: Basic Steps and Practice Problems

[Lee33]

Homework Statement

I was given this homework problem:

Show that if $a_1,a_2, ... ,$ is a sequence of real numbers that converges to $a$, then $$lim_{n\to \infty}\frac{\sum^n_{k=1} a_k}{n}=a.$$

I was provided a solution but my book never went over such examples or the concrete steps to solve such a problem. I am wondering what are the basic first steps to solving these types of problems?

And if possible, where can I find practice problems like these online? I searched but I couldn't find any.

 

[arildno]

Hint:
Remember that for every n, you may rewrite:
$$na=\sum_{k=1}^{n}a$$

 

[pasmith]

Homework Statement

I was given this homework problem:

Show that if $a_1,a_2, ... ,$ is a sequence of real numbers that converges to $a$, then $$lim_{n\to \infty}\frac{\sum^n_{k=1} a_k}{n}=a.$$

I was provided a solution but my book never went over such examples or the concrete steps to solve such a problem. I am wondering what are the basic first steps to solving these types of problems?

There are various techniques for limit problems, but since this problem asks you to start with an arbitrary convergent sequence $(a_k)$ the only one which will work is to use what you know about $(a_k)$: for all $\epsilon > 0$ there exists $K \in \mathbb{N}$ such that if $k \geq K$ then $a - \epsilon < a_k < a + \epsilon$.

That suggests taking an arbitrary $\epsilon > 0$ and its corresponding $K$ and splitting the sum as follows:
$$\frac1n \sum_{k=1}^n a_k = \frac1n \sum_{k=1}^{K-1} a_k + \frac1n \sum_{k=K}^n a_k$$
(You are interested in the limit $n \to \infty$, so at some stage you will have $n > K$ and you may as well assume that to start with.)

Your plan is to show that
$$a - \epsilon \leq \lim_{n \to \infty} \frac1n \sum_{k=1}^n a_k \leq a + \epsilon$$
and since $\epsilon > 0$ was arbitrary it must follow that
$$\lim_{n \to \infty} \frac1n \sum_{k=1}^n a_k = a$$
as required.

 

[mfb]

I would use ϵ/2 in one step instead of ϵ, that makes the inequalities easier to show.

 

[Lee33]

Thank you very much, pasmith! That cleared some issues I had, thanks for clarifying it for me.