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\begin{align}
\lim_{n \to \infty}\bracks{\pars{{\pi^{2} \over 6} - \sum_{k = 1}^{n}{1 \over k^{2}}}n} &=
\lim_{n \to \infty}\bracks{\pars{{1 \over n} -
2\int_{n}^{\infty}{\braces{x} \over x^{3}}\,\dd x}n}\label{1}\tag{1}
\end{align}
where we used a
well known identity which is related to the Riemann Zeta function.
Moreover,
$\ds{0 < 2\int_{n}^{\infty}{\braces{x} \over x^{3}}\,\dd x < {1 \over n^{2}}}$
such that \eqref{1} becomes:
$$\bbx{\ds{%
\lim_{n \to \infty}\bracks{\pars{{\pi^{2} \over 6} -
\sum_{k = 1}^{n}{1 \over k^{2}}}n} = 1}}
$$