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I (sort of) understand what Taylor series do, they approximate a function that is infinitely differentiable. Well, first of all, what does infinitely differentiable mean? Does it mean that the function has no point where the derivative is constant? Can someone intuitively explain that to me?

Anyway, so the function is infinitely differentiable, and the Taylor polynomial keeps adding terms which make the polynomial = to the function at some point, and then the derivative of the polynomial = to the derivative of the function at some point, and the second derivative, and so on.

Why does making the derivative, second derivative ... infinite derivative, of a polynomial and a function equal at some point ensure that the polynomial will match the function exactly?

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    "what does infinitely differentiable mean?" - if $f(x)$ is "infinitely differentiable", this means that if I differentiate $f(x)$ to obtain a new function $f^\prime(x)$, then I can differentiate $f^\prime(x)$ to obtain a new function $f^{\prime\prime}(x)$, which I can differentiate again to obtain... well, you get the drift. Additionally, all those derivatives should evaluate to finite values at the point of expansion.2012-08-15
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    Well how do I tell if a function is infinitely differentiable or not?2012-08-15
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    Do you have a calculator? (I guess WolframAlpha would work for this too.) Try playing around with taking the sine, cosine, and exponential (I mean $e^x$) of some very small numbers, ideally small powers of ten like $0.01, 0.001, 0.0001, ...$. You should notice some patterns. That's the Taylor series popping out at you.2012-08-15
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    @ordinary: Usually, you tell because you can express the function in terms of other functions you already know are infinitely differentiable and constructions you already know produce infinitely differentiable functions. For example, the sum of two infinitely differentiable functions is infinitely differentiable.2012-08-15

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