[tex] S_{1}= [/tex]1*2+2*3+...+n*(n+2)=∑n(n+2)=∑ n²+2∑n=[tex] \frac{n(n+1)(2n+1)}{6}+ 2\frac{n(n+1)}{2} [/tex]=[tex] \frac{n(n+1)(2n+7)}{6} [/tex]
∑n(n+1)=∑n²+∑n=[tex] S_{2}= \frac{n(n+1)(2n+4)}{6} [/tex]
[tex] \lim_{n \to \infty} \frac{ S_{1} }{ S_{2} }= \lim_{n \to \infty} \frac{n(n+1)(2n+7)}{n(n+1)(2n+4)}= \lim_{n \to \infty} \frac{2n+7}{2n+4}=1 [/tex]
