Derivative of Sum not Equal to Sum of Derivatives
One of my students complained this week after I took off points for assuming that a power series can be differentiated termbyterm with no further justification. He asked me for an example why this isn't ok.
I know of one example. The Taylor series for log(1+x) about x = 0 converges when x <= 1 except when x = 1. Evidently, this series is differentiable at x = 1 with derivative 1/(1+1) = 1/2. However, differentiating the series termbyterm and evaluating at x = 1 gives a series that does not converge.
I'm looking for further examples. More specifically:
1) Is there an example that only involves positive terms?
2) Is there an example where differentiating the series termbyterm and evaluating at a specific point gives a convergent series that converges to the wrong value (i.e. the derivative of the infinite fails to exist at the point or is has a different value)?
Of course, I know these things can only happen on the boundary of the disk of convergence
Thanks
