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Does the Taylor series of an infinitely differentiable function converge; and if yes, does it converge to the function? [duplicate]

I have googled it, but I am not satisfied with those. So my questions are: Let $D$ be an open set in $\mathbb{R}$. Let $f:D\rightarrow \mathbb{R}$ be a infinitely differentiable ...
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Prove Borel's Lemma (Pugh's book #35) [duplicate]

Given any sequence whatsoever of real numbers (a_r), there is a smooth function $f: \mathbb{R} \to \mathbb{R}$ such that $f^{(r)}(0) = a_r$. Pugh's hint says to try $f=\sum \beta_k(x)a_kx^k/k!$, ...
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Can the sequence of derivatives $\{f^{(k)}(0)\}_{k\geq 1}$ be any sequence? [duplicate]

Let $\{a_{k}\}_{k\geq 1}$ be any sequence of real numbers, must there exist a smooth function $f:]-\epsilon,\epsilon[\rightarrow \mathbb{R}$ (for some positive $\epsilon$) such that for every positive ...
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Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
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Are Taylor series and power series the same “thing”?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
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How to prove that a smooth function is NOT analytic?

For class we are exploring an example of a function that is smooth at $x=0$ but not analytic in any open interval centered at $0$. My question is, how can one prove that a function is not analytic? I ...
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Is there a $C^\infty$ function $f$ s. t. the convergent radius of Maclaurin series of $f$ is $0$?
To be precisely, let $f$ be a $C^\infty$ function defined on $(-\epsilon,\epsilon)$, where $\epsilon>0$. Question: Can the power series $\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}x^n$ have convergent ...
Is $C^\infty_p$ a finite dimensional vector space or infinite dimensional, for any point $p$ in a manifold $M$? The notation means the space of smooth germs.