my professor told me any n times differentiable function can be approximated by macularine/taylor expansion.(adsbygoogle = window.adsbygoogle || []).push({});

is that true? As far as I know, if the function is approximated at point a, the approximation is valid if we pick a point near a.

however, if we assume that we picked a point of a+infinite=b, how do we know the approximation can we conclude the approximation is valid or not?

For example, if we approximate a function with macularine series, and picked the point b=infinite for x, shouldn't we check the ratio test first?

ex.lim_{a,n→∞}[ƒ^{(n+1)}(a)*(x-a)^{(n+1)}/(n+1)!]/[ƒ^{(n)}(a)*(x-a)^{(n)}/(n)!]

simplyfying this equation gives

lim_{a,n→∞}[ƒ^{n+1}(a)*(x^{n+1}+ .... +a^{n+1})]/[ƒ^{n}(a)*(n+1)*(x^{n}....+a^{n}]

if it converges for every continuous smooth n time differetiable function, can anyone provie me proof?[/SUB]

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# Maclaurin expansion and error

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