Infinite Series: rearrangement of Terms

[MrBailey]

Hello,
I could use a big helping hand in trying to understand an example from a text.
Let's say I have a convergent series:

$$S=\sum_{n=1}^{\infty}{a_n}$$

Okay, so now:

$$\sum_{n=1}^{\infty}{a'_n}$$

is a rearrangement of the series where no term has been moved more than 2 places.

So, the exercise is to show that the rearranged series has the same sum and is also convergent.

A partial sum of the original series:

$$S_N=a_1+a_2+\ldots +a_N = S_{N-2}+a_{N-1}+a_N$$

A partial sum of the rearranged series:

$$S'_N=a'_1+a'_2+\ldots +a'_N = S'_{N-2}+a'_{N-1}+a'_N$$

I think I need to somehow bound
$$S'_N$$
but I'm not sure how.

Can someone steer me in the right direction?
Thanks,
Bailey

 

[shmoe]

Try to bound $$|S'_N-S_N|$$

 

[MrBailey]

Okay, is this on the right track?

So,

$$S_n = a_1 + a_2 + \ldots + a_n$$

and

$$S'_n$$ is a rearrangement of the series where the terms move no more than 2 places.

Then

$$|S'_n - S_n| \leq |a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|$$

where $$n < n_1 < n_2$$

From that we get:

$$S_n - (|a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|) \leq S'_n \leq S_n + (|a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|)$$

but since the original series converges: $$a_n \rightarrow 0$$

Therefore, $$S_n \leq S'_n \leq S_n$$ or $$S_n = S'_n$$

Is this correct?

How do I show that the new series converges as well?
Thanks,
Bailey

 

[shmoe]

Therefore, $$S_n \leq S'_n \leq S_n$$ or $$S_n = S'_n$$

What values of n do you think this equation is true?