The binomial theorem states that for an integer exponent n in NN:
(a+x)^n = sum_(k=0)^n ((n),(k)) x^ka^(n-k)
where ((n),(k)) is the binomial coefficient:
((n),(k)) = (n!)/(k!(n-k)!)
When the exponent is a generic real number nu notin NN we can introduce the generalized binomial coefficient:
((nu ),(k)) = (nu * ( nu -1) ( nu - 2 ) * ... * (nu - k+1))/(k!)
Now developing the function f(x) = (a+x)^nu in MacLaurin series we have:
f(0) = a^nu
f'(x) = nu (a+x)^(nu-1) so f'(0) = nu*a^(nu-1)
f''(x) = nu(nu -1) (a+x)^(nu-2) so f''(0) = nu(nu -1)a^(nu-2)
and we can easily see that in general:
f^((n))(0) = nu (nu - 1) (nu -2)* ... (nu -n +1) a^(nu-n)
so that the MacLaurin series is:
(a+x)^nu = sum_(k=0)^oo (nu (nu - 1) (nu -2)* ... (nu -k +1))/(k!) a^(nu-k) x^k
that is:
(a+x)^nu = sum_(k=0)^oo((nu),(k)) x^ka^(nu-k)
Using the MacLaurin theorem we can state that if we truncate the series at the n-th term, the rest is an infinitesimal of order higher then x^n for x->0:
(a+x)^nu = sum_(k=0)^n((nu),(k)) x^ka^(nu-k)+o(x^n)
which can be seen as a generalization of the binomial formula.