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Is the radius of convergence of a power series always equal to the radius of convergence of it's primitive or when its differentiated?

I have done a few examples and have noticed this. I am trying to understand this graphically and what i have been able to interpret is that when a graph is differentiable at a certain interval (the radius of convergence), it's differential will also exist at that interval. Is this correct? or is there more to it?

Thanks in advance.