Name of a Sum differentiation Trick

Question

in the proof here, they derive to make the sum prettier. Does this trick have a name and what is its general use case?

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Answer 1

Tim Gowers has an article in his blog called Differentiating Power Series where he says

I want to discuss the result that a power series $\sum_{n=0}^\infty a_nz^n $ is differentiable inside its circle of convergence, and the derivative is given by the obvious formula $\sum_{n=1}^\infty na_nz^{n-1}$.

This was part of his Analysis IA course in 2014, i.e. a first year Mathematics course at the University of Cambridge.

In effect it allows you to move the differential operator across the summation, thus providing a tool for evaluating some series. It is particularly useful for generating functions.