Question: I have homework which is about CFGs, their simplification, and their normalized forms. I have also seen some examples on the internet, but unfortunately, I could not solve the below question.

> 
> All the binary numbers, in which the $i$th character is equal to the
> character which is located in $i + 2$ th position, and the length of
> these strings is at least $2$.
> 
> 
> 

My problem is with positions, and how to show them using our grammar.
My idea was to have $S$ as our initial state, and then have a production like this:

$S\Rightarrow A|B$
------------------

And then $A$ be for all the strings which start by $0$, and $B$ be for all the strings which start by $1$.
But I will be really grateful for your help.

Answer: 
The subsequence $\left((-1)^{2n}\frac{2n}{2n+1}\right)\_{n\in\Bbb N}$ of the sequence$$\left((-1)^n\frac n{n+1}\right)\_{n\in\Bbb N}\tag1$$ is the sequence $\left(\frac{2n}{2n+1}\right)\_{n\in\Bbb N}$ whose limit is $1$. Since the sequence $(1)$ has a subsequence which converges to a number different from $0$, it does not converge to $0$, and therefore your series diverges.
You can argue like this: If $(a\_n)\_{n}$ is not a nullsequence also $(-1)^n a\_n$ is not a null sequence and hence the sequence is diverging.