The first few terms of the Maclaurin series can be obtained explicitly:
$$1-x-{\frac { \left( d-1 \right) \left( K-1 \right) }{2\,Kd+2}}{x}^{2 }-{\frac { \left( d-1 \right) \left( d-2 \right) \left( K-1 \right) \left( K-2 \right) }{ \left( 6\,Kd+6 \right) \left( Kd+2 \right) }}{ x}^{3}+O \left( {x}^{4} \right) $$