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\(\displaystyle \lim_{n\to\infty} \frac{1}{n^3}\left[n^2+\left(n+1\right)^2+\cdots\left(3n-1\right)^2\right]\)

\(\displaystyle =\lim_{n\to\infty} \sum_{k=0}^{2n-1} \left(1+\frac{k}{n}\right)^2\cdot\frac{1}{n}\)

\(\displaystyle =\int_1^3 x^2 dx\)

\(\displaystyle =\frac{x^3}{3}\Bigg|_{x=1}^{x=3}\)

\(\displaystyle =\frac{26}{3}\)

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