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Evaluate the infinite continued fraction $$\xi = [1,2,3,4,5\cdots] = 1+\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\vdots}}}$$

There is no periodic behavior of this continued fraction, I am not even sure that $\displaystyle\lim_{n \to \infty} [1,2,3,\cdots,n]$ exists or not. Any hints on how to solve this problem?

Sam
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    Could a continued fraction with 1 in all numerators and all entries strictly positive ever fail to converge? – Arthur Nov 06 '19 at 09:05
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    https://math.stackexchange.com/questions/1681993/why-is-1-frac11-frac11-ldots-not-real – Sam Nov 06 '19 at 09:07
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    See also https://math.stackexchange.com/questions/1871733/convergence-of-a-harmonic-continued-fraction. – Martin R Nov 06 '19 at 09:08
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    @Sabhrant That's not strictly positive entries. The numerators are all $-1$. – Arthur Nov 06 '19 at 09:08

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