Cantor's Serpent

Calculus Level 5

It is well known that one can create a bijection between the natural numbers and the rational numbers by snaking diagonally through the entries of a grid, as can be seen in the picture.

We can define a sequence \(\lbrace a_{i} \rbrace_{i=1}^{\infty}\) in that manner, ie

\[\begin{array}{ccc} a_1 = \frac{1}{1} & a_2 = \frac{2}{1} & a_3 = \frac{1}{2} \\ a_4 = \frac{1}{3} & a_5 = \frac{2}{2} & a_6 = \frac{3}{1} \\ a_7 = \frac{4}{1} & a_8 = \frac{3}{2} & \ldots \end{array}\]

Find the value of

\[\displaystyle \lim_{n \rightarrow \infty} \frac{\mu(a_{i}, n(n+1)/2)}{n \ln n}\]

where \(\mu(x_{i}, n)\) denotes the mean of the first \(n\) elements of \(\lbrace x_{i} \rbrace\).


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