The value of $$\sum\limits_{n = 0}^\infty {n{{\left( {{1 \over 2}} \right)}^n}\,\,} $$ is _______.

The contour on the $$x-y$$ plane, where the partial derivative of $${x^2} + {y^2}$$ with respect to $$y$$ is equal to the partial derivative of $$6y+4x$$ with respect to $$'x',$$ is

The series $$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$ converges to

The maximum value of the function $$\,f\left( x \right) = \ln \left( {1 + x} \right) - x$$ (where $$x > - 1$$ ) occurs at $$x=$$________.