# 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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