Repeated Differentiation Leading to Tetrationally Fast Growth

[Catullus]

Does anyone know of a function , such that  grows like , for a real a, and for a fixed k?

[Daniel]

Does anyone know of a function , such that the xth derivative of  grows tetrationally fast, for a fixed k?

First I should mention that the function you are looking for would be wildly divergent. While I don't know of any publication of such a function, it's construction would be straight forward.

For a given \( a\in\mathbb R \),
\( \sum_{n=0}^\infty ^na x^n \)

[MphLee]

Does anyone know of a function , such that the xth derivative of  grows tetrationally fast, for a fixed k?

First I should mention that the function you are looking for would be wildly divergent. While I don't know of any publication of such a function, it's construction would be straight forward.

For a given \( a\in\mathbb R \),
\( \sum_{n=0}^\infty ^na x^n \)

Hi Daniel, I was thinking the same but then... what the kth derivatives of n-hyperexponentiation would look like?
Let \(h_n\) be hyperexponentiation of rank \(n\) of base \(b\) fixed. What about
\[{\mathcal H}_{n,x}(k)=\frac{d^k}{d^kx}h_n(x)\]

Does exist an \(n\) s.t. \(h_4(k)\leq{\mathcal H}_{n,x}(k)\)?

I wonder if this identity is telling us something \(\frac{h'_{n+1}(x+1)}{h'_{n+1}(x)}=h'_{n}(h_{n+1}(x))\)

[JmsNxn]

Does anyone know of a function , such that the xth derivative of  grows tetrationally fast, for a fixed k?

First I should mention that the function you are looking for would be wildly divergent. While I don't know of any publication of such a function, it's construction would be straight forward.

For a given \( a\in\mathbb R \),
\( \sum_{n=0}^\infty ^na x^n \)

Hi Daniel, I was thinking the same but then... what the kth derivatives of n-hyperexponentiation would look like?
Let \(h_n\) be hyperexponentiation of rank \(n\) of base \(b\) fixed. What about
\[{\mathcal H}_{n,x}(k)=\frac{d^k}{d^kx}h_n(x)\]

Does exist an \(n\) s.t. \(h_4(k)\leq{\mathcal H}_{n,x}(k)\)?

I wonder if this identity is telling us something \(\frac{h'_{n+1}(x+1)}{h'_{n+1}(x)}=h'_{n}(h_{n+1}(x))\)

No, Daniel's right, there is no such function--at least no such analytic function. I doubt there's even a smooth function.

Every analytic function must satisfy:

\[
\lim_{n\to\infty} \left(\frac{f^{(n)}(k)}{n!}\right)^{1/n} \le 1
\]

And it's safe to say:

\[
\lim_{n\to\infty} \left(\frac{^n a}{n!}\right)^{1/n} = \infty\\
\]

[Catullus]

No, Daniel's right, there is no such function--at least no such analytic function. I doubt there's even a smooth function.

Every analytic function must satisfy:

\[
\lim_{n\to\infty} \left(\frac{f^{(n)}(k)}{n!}\right)^{1/n} \le 1
\]

And it's safe to say:

\[
\lim_{n\to\infty} \left(\frac{^n a}{n!}\right)^{1/n} = \infty\\
\]

 How many times real differentiable do you think it could be?
Do you have any more guesses about that?

[tommy1729]

No, Daniel's right, there is no such function--at least no such analytic function. I doubt there's even a smooth function.

Every analytic function must satisfy:

\[
\lim_{n\to\infty} \left(\frac{f^{(n)}(k)}{n!}\right)^{1/n} \le 1
\]

And it's safe to say:

\[
\lim_{n\to\infty} \left(\frac{^n a}{n!}\right)^{1/n} = \infty\\
\]

How many times differentiable do you think it could be?
Do you have any more guesses about that?

Radius zero implies not defined outsite its expansion point.

So nothing to take the derivative from.

unless you have another nontaylor definition that does compute values ofcourse.

regards

tommy1729

ps : a taylor expansion expanded at another point will not work since your derivatives diverges faster than exponential.

to see this : if f(x) has radius zero and derivatives diverges faster than exponential then f( c x ) has the same problem.

Plz think about it before you reply.

regards

tommy1729