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 convergence disk |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) = α u(x) with initial data u(0) = 1. The unique solution of this problem is the function u(x) = (1 + x)α, which is therefore the sum of the binomial series, at least 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)α.
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