$$\eqalign{ & \cfrac{1}{{1 + x}} + \cfrac{{\cfrac{1}{{1 + x/2}} + \cfrac{{\cfrac{1}{{1 + x/3}} + \cfrac{{...}}{{3/x}}}}{{2/x}}}}{{1/x}} = \cr & = x \cdot {\left( { - x} \right)^{ - x}} \cdot \left( {\Gamma \left( x \right) - \Gamma \left( {x, - x} \right)} \right) \cr & = x \cdot {\left( { - x} \right)^{ - x}}\int\limits_0^{ - x} {{t^{x - 1}}{e^{ - t}}dt} = x{\left( { - x} \right)^{ - x}}\gamma \left( {x, - x} \right) \cr} $$