<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] Notice that for $n\ge1$ you have
$$\frac{n^3+n^2+n}{n^3+1}=
1+\frac{n^2+n-1}{n^3+1} \ge
1+ \frac{n^2-n+1}{n^3+1} = 
1+\frac1{n+1} = \frac{n+2}{n+1}.$$
Therefore
$$\prod_{k=1}^n \frac{k^3+k^2+k}{k^3+1} \ge \prod_{k=1}^n \frac{k+2}{k+1} = \frac32 \cdot \frac 43 \cdots \frac{n+2}{n+1} = \frac{n+2}2.$$
And for $n\to\infty$ we have $\lim\limits_{n\to\infty} \frac{n+2}2=\infty$.
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