term-by-term differentiation and integration
Differentiating an infinite sum looks dangerous: a derivative is itself a limit, and a sum is another limit, and limits do not always commute. But inside the disk of convergence of a power series, this danger evaporates entirely. You may differentiate the series exactly as if it were a finite polynomial — differentiate each term separately and add the results — and you get the honest derivative of the function the series defines. The same freedom holds for integration along a path. This is one of the most pleasant facts in the subject.
Concretely, if f(z) = sum a_n (z - z_0)^n for |z - z_0| < R, then f'(z) = sum n a_n (z - z_0)^(n-1), obtained by differentiating each term, and this new series has the SAME radius of convergence R. Likewise an antiderivative is sum a_n (z - z_0)^(n+1) / (n+1), again with radius R. Because the radius is unchanged, you can repeat the process forever: f has derivatives of every order inside the disk, so a power-series function is automatically infinitely differentiable. The justification is the uniform convergence on compact subsets — that is the property that lets you legally interchange the derivative (or the integral) with the infinite sum.
The payoff is enormous. First, it shows a power series defines a holomorphic function, since it is complex-differentiable everywhere in its disk. Second, differentiating the series n times and setting z = z_0 isolates the coefficient: a_n = f^(n)(z_0) / n!, the Taylor formula. So the coefficients are not arbitrary numbers; they are the derivatives of the function, neatly packaged. Third, term-by-term integration lets you build new series from old ones — for instance integrating sum z^n = 1/(1 - z) term by term gives the series for -Log(1 - z) for free.
Start from sum z^n = 1/(1 - z) on |z| < 1. Differentiate term by term: sum n z^(n-1) = 1/(1 - z)^2, still on |z| < 1. Integrate the original term by term from 0: sum z^(n+1)/(n+1) = -Log(1 - z), again radius 1. Three useful series, all squeezed out of one by legal termwise operations.
Differentiating and integrating a power series never changes its radius of convergence, so the whole family of related series shares one disk.
Term-by-term operations are legal only inside the open disk, where convergence is uniform on compacta — never assume you may also differentiate term by term at points on the boundary circle, where the differentiated series can diverge.