# Proving that this number is irrational

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?