How to symbolically manipulate the closed form series output from the easyFourier package?

EasyFourier by @xzczd is a nice package to obtain a Fourier series in closed form, e.g.

f = x^2 easyFourierTrigSeries[f, {x, -\[Pi], \[Pi]}, \[Infinity]] 

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However, the output cannot be manipulated symbolically, at least I don’t know how to. For instance, differentiating the Fourier series is not possible like this:

D[easyFourierTrigSeries[f, {x, -\[Pi], \[Pi]}, \[Infinity]],x] 

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What I would like to do is something comparable to the following, where I have copied the summand manually into a regular Mathematica Sum:

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BTW: How do I downscale the size of pictures in stack exchange? Is there no easier way to insert Mathematica output? How can I insert complicated Mathematica input code in 2d-mode? (Thanks for hints to these "meta" questions, too.)