Understanding Radius of Convergence in Power Series: A Graphical Approach

[physicsnoob93]

Hi. Not really a homework question. Just a doubt i would like to confirm.

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.

 

[HallsofIvy]

Not necessarily. If a power series converges, not only inside the radius of converges but also at the end points, then it converges uniformly and so the series formed by differentiating term by term or integrating term by term must also converge on that same interval. However, if a power series does not converge at one or both end points, then it does NOT converge uniformly within the radius of convergence and the radius of convergence of its term by term derivative or term by term integral may be smaller than the radius of convergence of the original function.