Linked Questions

2
votes
3answers
2k views

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 ...
1
vote
1answer
436 views

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 $f=\sum \beta_k(x)a_kx^k/k!$, ...
3
votes
1answer
73 views

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 ...
0
votes
0answers
31 views

Sequence of derivatives [duplicate]

Is it true that every sequence of real numbers $(a_n)$ can be a sequence of derivatives - ($f^{(n)}(0)$) of some function $f\in C^{\infty}(\mathbb{R})$? It's clear if the series $\sum \frac{a_n}{n!}x^...
2
votes
0answers
26 views

Let $\{a_n\}\in\mathbb R^{\mathbb N}$. Does there exist $h\in C^\infty(\mathbb R)$ with $h^{(n)}(0)=a_n$ for every $n$? [duplicate]

If the power series $\sum_{n\in\mathbb N}a_n\frac{x^n}{n!}$ converges for all real $x$, then the answer is trivially yes. If the power series above has radius of convergence $3r>0$, let $\psi_r\in ...
58
votes
16answers
4k views

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 ...
16
votes
4answers
30k views

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 ...
7
votes
2answers
2k views

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 ...
10
votes
2answers
858 views

Every real sequence is the derivative sequence of some function

I am looking for the proof of the following theorem: Let $(a_n)$ be a sequence of real numbers. Then there exists a function $f$ which is infinitely differentiable at 0, and $$ \frac{d^nf}{...
23
votes
1answer
931 views

A curious theorem by Peano

Let $f$ be defined on $[a,b]$ and there differentiable. Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac {f(a_{...
10
votes
1answer
584 views

Existence of smooth extension of a function defined on a closed interval

Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
4
votes
1answer
2k views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
4
votes
2answers
436 views

Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
3
votes
1answer
94 views

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 ...
1
vote
1answer
207 views

Is the space of smooth germs on a manifold finite dimensional or infinite-dimensional?

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.

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