View Single Post 2008-11-22, 07:57 #1 jinydu   Dec 2003 Hopefully Near M48 2×3×293 Posts 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 term-by-term 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 term-by-term 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 term-by-term 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  