Repeated Differentiation Leading to Tetrationally Fast Growth

[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