nt.number theory - Irrational logs and the harmonic series

Consider the series
$$S_f = sum_{x=1}^infty frac{f}{x^2+fx}.$$
Goldbach showed that, for integers $f ge 1$,
$$S_f = 1 + frac12 + frac13 + ldots + frac1f$$
(this follows easily by writing $S_f$ as a telescoping series).
Thus $S_f$ is rational for all natural numbers $f ge 1$.
Goldbach claimed that, for all nonintegral (rational) numbers $f$,
the sum $S_f$ would be irrational.

Euler showed, by using the substitution
$$frac1k = int_0^1 x^{k-1} dx,$$
that
$$S_f = int_0^1 frac{1-x^f}{1-x} dx.$$
He evaluated this integral for $f = frac12$ and found
that $S_{1/2} = 2(1 - ln 2)$ (this also follows easily from
Goldbach's series for $S_f$). Thus Goldbach's claim holds for all
$f equiv frac12 bmod 1$ since $S_{f+1} = S_f + frac1{f+1}$.

Here are my questions:

1. The irrationality of $ln 2$ was established by Lambert, who
   proved that $e^r$ is irrational for all rational numbers
   $r ne 0$. Are there any (simple) direct proofs?




2. Has Goldbach's claim about the irrationality of $S_f$ for
   nonintegral rational values of $f$ been settled in other cases?

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