Keep calm and do the limit

Algebra Level 5

\[\large\dfrac{\displaystyle\sum_{k=1}^{n-1}x_kx_{k+1}}{\displaystyle\sum_{k=1}^{n}x_k^2}\]

Let \(M_n\) be the maximum of the expression above, where \(x_1,\ldots,x_n\) are positive real numbers. Find \(\displaystyle \lim_{n\to\infty}M_n\).

Bonus question: Find a closed formula for \(M_n\).

Note: The limit is much easier to find than \(M_n\) itself.


Related problems here and here.

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