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Old 2008-11-22, 07:57   #1
jinydu's Avatar
Dec 2003
Hopefully Near M48

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Default 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

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