Keep calm and do the limit

Algebra Level 5

k=1n1xkxk+1k=1nxk2\large\dfrac{\displaystyle\sum_{k=1}^{n-1}x_kx_{k+1}}{\displaystyle\sum_{k=1}^{n}x_k^2}

Let MnM_n be the maximum of the expression above, where x1,,xnx_1,\ldots,x_n are positive real numbers. Find limnMn\displaystyle \lim_{n\to\infty}M_n.

Bonus question: Find a closed formula for MnM_n.

Note: The limit is much easier to find than MnM_n itself.


Related problems here and here.

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