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September 12th, 2017, 05:41 PM   #1
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Question Rules for differentiating series?

I'm not very clear on the rules for differentiating series. Can series only be differentiated under certain circumstances? If so, what are they? Must the series be infinite?

For example, suppose that I know the value of a series, whether the series is finite or infinite, is equivalent to a given expression dependent upon a variable A.

If I differentiate that expression, can I always just differentiate the series with respect to A? Or are there are there situations where that won't work?
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September 12th, 2017, 06:24 PM   #2
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Just differentiate term by term.

If you know that a series converges to some function of a variable, say, $f(t)$

and if you also know analytically that $\dfrac{df}{dt}$ exists,

then the term by term differentiation of that series will converge to $\dfrac{df}{dt}$.

If the series doesn't converge, I don't think you can say anything immediately about the convergence of the derivative.

The series of $\dfrac{1}{n}$ doesn't converge, but its derivative $-\dfrac{1}{n^2}$ does.

Last edited by skipjack; September 13th, 2017 at 11:23 AM.
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September 13th, 2017, 04:05 AM   #3
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The only "rule" is that the series must converge uniformly in order to be sure that the series, differentiated term by term, will converge to the derivative of the original function. That is, if $\displaystyle \sum_{n=0}^\infty f_n(x)$ converges uniformly to f(x) then $\displaystyle \sum_{n=0}^\infty f'_n(x)$ converges to f'(x).
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Last edited by skipjack; September 13th, 2017 at 11:27 AM.
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