Convergence of \sum_{n=0}^{\infty} zeta^{(n)}(s)

[seanhbailey]

Homework Statement

Does the sum $$\sum_{n=0}^{\infty} zeta^{(n)}(s)$$ converge regardless of s, where n is the nth derivative of the Riemann Zeta function? If it converges tell what value it converges to.

Homework Equations

The Attempt at a Solution

I used the integral test, and I think it diverges. Plus, by plotting the sequence on Matematica, it looks like it is diverging. Am I correct?

 

[Count Iblis]

Divergent, but you can still resum the series using Borel resummation.

 

[seanhbailey]

How do i do that?

 

[seanhbailey]

Can anyone prove that the sum $$\sum_{n=0}^{\infty} zeta^{(n)}(s)$$ goes to INFINITY not just diverges

 

[Count Iblis]

Can anyone prove that the sum $$\sum_{n=0}^{\infty} zeta^{(n)}(s)$$ goes to INFINITY not just diverges

Apply the operator 1 - d/ds to the summation. To investigate convergence, you can apply it to the partial sums of the first n terms. If you apply it formally to the infinite summation then you get the same result you would get after performing the Borel resummation.