information:filtering
Summation of the binomial series

The usual argument to compute the sum of the binomial series goes as follows. Differentiating term-wise the binomial series within the disk of convergence | x | < 1 and using formula (1), one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x) u ′(x) − αu (x) = 0 with initial condition u (0) = 1.

The unique solution of this problem is the function u (x) = (1 + x). Indeed, multiplying by the integrating factor (1 + x) gives

so the function (1 + x) u (x) is a constant, which the initial condition tells us is 1. That is, u (x) = (1 + x) is the sum of the binomial series for | x | < 1.

The equality extends to | x | = 1 whenever the series converges, as a consequence of Abel's theorem and by continuity of (1 + x).