<s>[INST] Why does $\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverge? $\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverges, and I have no idea why? It would seem using L'hop, $\frac{n^3 + n^2 + n}{n^3 + 1}$ goes to 1. So it should end up just being $1\cdot1\cdot1\cdots$, which makes me feel it converges. Is the problem that it always adds that $.000\ldots0001$?
 [/INST] Intuitively, as $\frac{n^3 + n^2 + n}{n^3 + 1} \sim 1$, if we erase $n$ from the top and $1$ from the bottom, the fraction should get smaller. Lets prove it:
$$\frac{n^3 + n^2 + n}{n^3 + 1}-\frac{n+1}{n}=\frac{n^2-n-1}{n(n^3+1)}$$
Thus, for $n \geq 2$ we have
$$\frac{n^3 + n^2 + n}{n^3 + 1} > \frac{n+1}{n}$$
Now, since it is telescopic:
$$\prod_{n=1}^N \frac{n+1}{n} =N+1$$
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