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The examples given here for example, show that once you know the form of a taylor polynomial as a function of $x$, you can replace the $x$ with another function. It works when you work out the problems but I don't understand why. It would seem that the chain rule would make the taylor polynomial something else.

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Could you please be more specific about what in that link seems unclear? It may be easier to explain with an example in mind. – Jonas Meyer Jan 27 '11 at 2:27
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If $\displaystyle{f(x)=\sum_{n=0}^\infty a_n(x-c)^n}$ is a Taylor series that converges in the interval $(c-R,c+R)$, and if $g$ is a function, then $\displaystyle{f(g(x))=\sum_{n=0}^\infty a_n(g(x)-c)^n}$ is valid as long as $g(x)$ is in the interval $(c-R,c+R)$. $f(y)=\sum\limits_{n=0}^\infty a_n(y-c)^n$ and $f(t)=\sum\limits_{n=0}^\infty a_n(t-c)^n$ converge if $y$ and $t$ are in $(c-R,c+R)$. In the last statement, let $t=g(x)$. The point is that whatever you plug in in place of the original "$x$", the series converges as long as that thing is in $(c-R,c+R)$. I don't know if that helps. – Jonas Meyer Dec 13 '11 at 5:59
The new series isn't necessarily a Taylor series. You still get a series, but the terms aren't (necessarily) of the form $c_n (x-x_0)^n$ (e.g. the substitution $x\mapsto x^2-1$). The only cases where you do get a Taylor series back out are linear substitutions $x\mapsto ax-b$, in which case it's straightforward to check that the resulting series is consistent with the true Taylor series. – Riley E Dec 13 '11 at 8:36
(My above comment is only in regards to the "It would seem that the chain rule would make the taylor polynomial something else." part of the question) – Riley E Dec 13 '11 at 8:46

closed as not a real question by Jonas Meyer, t.b., Jonas Teuwen, Srivatsan, J. M. Dec 13 '11 at 10:35

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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See this (theorem concerning substitution in Taylor series).

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