Binomial coefficients C (n, k) extended for negative and fractional n, illustrated with expansion of a simple binomial by CMG Lee. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case n = −1 gives Grandi's series.

${\displaystyle {\sqrt {1+x}}}$

${\displaystyle x\rightarrow -x^{2}}$

${\displaystyle {\tfrac {\pi }{12}}+{\tfrac {\sqrt {3}}{8}}}$

${\displaystyle \pi }$