<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] The numerator is bounded above by $n.n(n+1)(n+2)$, fourth degree. The denominator is exactly $(2n^3+3n^2+n)^2/36$ (from the square pyramidal number formula), sixth degree. So the denominator "wins".
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