Why does this sequence not oscillate?

Question

I'm confused on why this sequence converges to 0 rather than diverge because of an oscillating series, since the result would be negative if n is odd and positive if n is even

Edit for the downvote: I apologize that my question was simple, but I genuinely thought it through for a while and could not understand. I was hoping I could ask the smart and giving people of Stack Exchange in order to enlighten me.

Answer 1

Notice we have: If $|a_n| \to 0$, then $a_n \to 0$ because of

$$ - |a_n| \leq a_n \leq |a_n| $$

and the squeeze rule

Answer 2

For $n>0$, we have

$$a_{2n}=\frac{2n}{8n^3+1}>0$$

and

$$a_{2n+1}=-\frac{2n+1}{(2n+1)^3+1}<0$$

thus

$$\lim_{n\to+\infty}a_{2n}=0$$

and

$$\lim_{n\to+\infty}a_{2n+1}=0$$

So,

$$\lim_{n\to+\infty}a_n=0.$$