Die Eulersche Zahl als Kettenbruch

$$\begin{array}{lrlr}& \mathrm{e}& =2+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{2+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{4+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{6+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{8+{\displaystyle \frac{1}{\ddots }}}}}}}}}}}}}}}}}}}}}}}}& \text{<span class="tml-eqn"></span>}\\ & & =[2;\mathrm{1,2,1,1,4,1,1,6,1,1,8},\dots ]& \text{<span class="tml-eqn"></span>}\end{array}$$