The following Engel expansion can be simplified: $\frac{a^0}{2}+\frac{a^1}{2^2}+\frac{a^2}{2^3}+\frac{a^3}{2^4}+\ldots+\frac{a^{n-1}+a^nx}{2^n}=\frac{a^{n-1}(x+1)-2^{n-1}}{a^{n-1}}$
How do we simplyfiy the expansion:
$\frac{a^0}{2}+\frac{a^1}{2\cdot4}+\frac{a^2}{2\cdot4\cdot2}+\frac{a^3}{2\cdot4\cdot2\cdot4}+\ldots+\frac{a^{n-1}+a^nx}{2\cdot4\cdot2\cdot4\cdot2\cdot4\ldots}$
or even
$\frac{a^0}{2}+\frac{a^1}{2\cdot2}+\frac{a^2}{2\cdot2\cdot4}+\frac{a^3}{2\cdot2\cdot4\cdot2}+\ldots+\frac{a^{n-1}+a^nx}{2\cdot2\cdot4\cdot2\cdot2\cdot4\ldots}$
It would be very interesting to generalize such behavior.