How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$?

That is,

How to evaluate

$\frac{1}{\left ( \frac{1}{1} \right )+\frac{1}{\left ( \frac{3}{2} \right)+\frac{1}{\left ( \frac{11}{6} \right )+\frac{1}{\left ( \frac{25}{12}\right )+\frac{1}{\ddots}}}}}$?

The value of the expression is $0.6606\dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?

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    $\begingroup$ Probably, the value of this (unusual) continued fraction does not have a "closed-form-representation" $\endgroup$ – Peter Dec 7 '18 at 9:20
  • $\begingroup$ @Peter Probably, it does have a closed-form-representation, because $a_{j}$ has a formula to be evaluated, (where this formula contains the Euler's constant, $\gamma$). I think, since $a_{j}$ has a formula, then the given expression has a closed-form-representation. $\endgroup$ – Hussain-Alqatari Dec 7 '18 at 9:26
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    $\begingroup$ @Hussain-Alqatari What you call a formula is just an approximation. Even if it was exact this would not imply that your expression has a closed form. $\endgroup$ – Christoph Dec 7 '18 at 9:31
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    $\begingroup$ Have you had any luck in determining the "partial fractions," so to speak, of the series, and finding a closed form of them? (Sort of like how summations have "partial sums," etc., I don't know the proper term.) We speak of fractions like this converging to a value when the limit of those partial fractions approaches something finite, so that'd be my first attempt at trying to find a closed-form expression for this. Even a trend in the approximations/values of these partial fractions could hint at notable behavior. $\endgroup$ – Eevee Trainer Dec 7 '18 at 9:33
  • $\begingroup$ The first 40 Digits of the continued fraction are: 0.66061703554133547787137090685771784456141074402642822344481672518418485119954 $\endgroup$ – Dr. Wolfgang Hintze Dec 8 '18 at 17:35

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