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derivative of limit function diverges from limit of derivatives
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(Example)
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For a function sequence, one cannot always change the order of taking limit and differentiating, i.e. it may well be $$\lim_{n\to\infty}\frac{d}{dx}f_n(x) \;\neq\; \frac{d}{dx}\lim_{n\to\infty}f_n(x),$$ even in the case that a sequence of continuous (and differentiable) functions converges uniformly; cf. Theorem 2 of the parent entry.
Example. The function sequence
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(1) |
provides an instance; we consider it on the interval $[-1,\,1]$ . It's a question of partial sum the converging geometric series $$\frac{x^3}{1\!+\!x^2}+\frac{x^3}{(1\!+\!x^2)^2}+\frac{x^3}{(1\!+\!x^2)^2}+\ldots$$ (although one cannot use Weierstrass' criterion of uniform convergence). Since the limit function is $$f(x) \;:=\; \lim_{n\to\infty}\left(x-\frac{x}{(1\!+\!x^2)^n}\right) \;=\; x \quad \forall x \in [-1,\,1],$$
we have $$\sup_{[-1,\,1]}|f_n(x)-f(x)| \;=\; \sup_{[-1,\,1]}\frac{|x|}{(1\!+\!x^2)^n} \,\longrightarrow 0 \quad \mbox{as}\;\; n \to \infty,$$ which means by Theorem 1 of the parent entry that the sequence (1) converges uniformly on the interval to the identity function. Further, the members of the sequence are continuous and differentiable. Furthermore, $$f_n'(x) \;=\; 1-\frac{1\!+(1\!-\!2n)x^2}{(1\!+\!x^2)^{n+1}},$$ whence $$\lim_{n\to\infty}f_n'(x) \;=\; 1 \quad (x \;\neq\; 0).$$ But in the point $x = 0$ we
have $$\lim_{n\to\infty}f_n'(0) \;=\; \lim_{n\to\infty}0 \;=\; 0,$$ which says that the limit of derivative sequence of (1) is discontinuous in the origin. Because $$f'(x) \;\equiv\; 1,$$ we may write $$\lim_{n\to\infty}\frac{d}{dx}f_n \;\neq\; \frac{d}{dx}\lim_{n\to\infty}f_n.$$
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"derivative of limit function diverges from limit of derivatives" is owned by pahio.
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See Also: growth of exponential function
| Other names: |
limit of derivatives diverges from derivative of limit function |
| Keywords: |
function sequence, uniform convergence |
This object's parent.
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Cross-references: origin, discontinuous, derivative, limit, point, members, identity function, limit function, Weierstrass criterion of uniform convergence, geometric series, partial sum, interval, theorem, converges uniformly, functions, differentiable, continuous, sequence, function sequence
This is version 4 of derivative of limit function diverges from limit of derivatives, born on 2009-08-24, modified 2009-09-08.
Object id is 11877, canonical name is DerivativeOfLimitFunctionDivergesFromLimitOfDerivatives.
Accessed 496 times total.
Classification:
| AMS MSC: | 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions) | | | 26A15 (Real functions :: Functions of one variable :: Continuity and related questions ) |
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Pending Errata and Addenda
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