<s>[INST] How can I evaluate $\lim_{n \to\infty}\left(1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)\right)/\left(1^2+2^2+3^2+\dots+n^2\right)^2$? How can I evaluate this limit? Give me a hint, please. 
$$\lim_{n \to\infty}\frac{1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)}{\left(1^2+2^2+3^2+\dots+n^2\right)^2}$$
 [/INST] Note that
$$(1^2+2^2+\ldots+n^2)^2\ge 1^4+2^4+\ldots+n^4$$
Now we can use Stolz–Cesàro theorem (kind of a discrete  l'Hôpital's rule):
$$\lim_{n\to\infty}\frac{1\cdot2\cdot3+2\cdot3\cdot4+\ldots+n(n+1)(n+2)}{1^4+2^4+\ldots+n^4}=\lim_{n\to\infty}\frac{n(n+1)(n+2)}{n^4}=0$$
And conclude the original limit was also $0$.
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