Example 12. Find $\lim _{n \rightarrow \infty} \frac{1^{2}+2^{2}+3^{2}+\ldots+n^{2}}{n^{3}}$.
Solution. Using the formula for the sum of squares of a natural number series

$$
1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}
$$

we get

$$
\lim _{n \rightarrow \infty} \frac{1^{2}+2^{2}+3^{2}+\ldots+n^{2}}{n^{3}}=\lim _{n \rightarrow \infty} \frac{n(n+1)(2 n+1)}{6 n^{3}}=
$$

$$
\begin{gathered}
=\frac{1}{6} \lim _{n \rightarrow-}\left(\frac{n+1}{n}\right)\left(\frac{2 n+1}{n}\right)=\frac{1}{6} \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right) \cdot \lim _{n \rightarrow \infty}\left(2+\frac{1}{n}\right)= \\
=\frac{1}{6} \cdot 1 \cdot 2=\frac{1}{3}
\end{gathered}
$$