Liouville's Constant is defined as follows:

$$\begin{align} L_{10} &= \sum^\infty_{n=1}10^{-n!} \\ &=0.\mathbf{1}\text{(no zeros)}\mathbf{1}\text{(3 zeros)}\mathbf{1}\text{(17 zeros)}\mathbf{1}\text{(95 zeros)}\mathbf{1}\text{(599 zeros)}\mathbf{1}\text{(4319 zeros)}\mathbf{1}\text{(35279 zeros)}\ldots \end{align}$$

Notice that $\forall~n\in\mathbb{Z}^{+}$, the $n$th and $(n+1)$th occurrences of $1$ are separated by $nn!-1$ zeros.

Now let $\Lambda$ be the constant formed by summing the reciprocals of the numbers of zeros between each subsequent occurrence of the digit $1$ in Liouville's Constant:

$$\begin{align} \displaystyle\Lambda &= \sum^\infty_{n=2}\frac{1}{nn!-1} \\ &= \frac{1}{3}+\frac{1}{17}+\frac{1}{95}+\frac{1}{599}+\frac{1}{4319}+\frac{1}{35279}+\frac{1}{322559}+\frac{1}{3265919}+\ldots \\ &= 0.40461594459112144123383699845748495376409240909536\ldots \end{align}$$

How can I (dis)prove that $\Lambda$ is irrational?


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