I would like to know how are encoded the realanalytic functions on the interval by the computers. When I think in a realanalytic function I just think in a composition of the ''typical'' analytic functions of every day, in all of this cases it is possible to find an ''exact''(because there is an explicit formula for the taylor expansion at any point) representation of these functions. I want also to ask if someone could give an example of a realanalytic function for which the representation in a computer is very bad behaves.
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$\begingroup$ Regarding computer representation, there are two distinct issues: whether you ask for numerical or symbolic computation. Please clarify what you are referring to. $\endgroup$ – Loïc Teyssier Feb 8 '13 at 13:00

$\begingroup$ I mean symbolic computation. $\endgroup$ – Umberto Feb 8 '13 at 13:06

1$\begingroup$ Then Taylor expansion is not your best friend there, since the sum contains infinitely many terms. In general a CAS consider only elementary analytic function : finite sum/product/composition/inverse of powers of x and exponential, from which you build what you call "typical everyday" functions. It uses its functional relations (e.g. $\exp′=\exp$ or $\cos^2+\sin^2=1$). When you add transcendents (e.g. Bessel functions) the CAS also uses the functional equations it satisfies to simplify expressions. Just like the everyday mathematician. I do believe the CAS never uses a Taylor expansion. $\endgroup$ – Loïc Teyssier Feb 8 '13 at 13:50
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The function $\sum \frac{x^n}{n^n}$ has been discusssed. Other choices for the denominator might also give convergent functions defined by power series whose behavior is complicated. Maybe that is relevant.