4.11 Binomial coefficients

Hence we obtain the binomial series $(*)\,$ by substituting the values for the derivatives into the Maclaurin series formula.

Note that, when $\alpha\,$ is a positive integer, $f(x)=(1+x)^{\alpha}$ has $f^{(n)}=0\,$ for $n>\alpha\,$; so the series terminates and gives a polynomial. Thus we recover the binomial expansion which we had before with

as in Pascal’s triangle. In this special case, the binomial coefficient is a positive integer, whereas this is not typical for the coefficients from the general binomial expansion.