I am really wondering whether one can something achieve with fractional differentiation with respect to tetration, sounds quite promising, however
this would mean that you can develop the logarithm at 0 into a powerseries, which is not possible.
I guess the problem in your derivations occurs after this line:
I assume this line is still convergent, however if you separate the difference into two sides you work with two divergent series.
Remember \( \lim_{n\to\infty} a_n - b_n = \lim_{n\to\infty} a_n - \lim_{n\to\infty} b_n \) *only* if all (or at least two of the three) limits exists.
(05/26/2011, 02:50 AM)JmsNxn Wrote: I decided to multiply the infinite series and I got:
\( ln(x) = \sum_{n=0}^{\infty} x^n (\sum_{k=0}^{n} (-1)^k \frac{\sum_{c=1}^{n-k}\frac{1}{c} - \gamma}{k!(n-k)!}) \)
Using Pari gp nothing seems to converge, but that may be fault to may coding.
this would mean that you can develop the logarithm at 0 into a powerseries, which is not possible.
I guess the problem in your derivations occurs after this line:
(05/26/2011, 02:50 AM)JmsNxn Wrote: \( 0 = \sum_{n=0}^{\infty} x^{n-t}\frac{\Gamma(n+1)}{n!\Gamma(n+1-t)}(\psi_0(n+1-t) - \ln(x)) \)
I assume this line is still convergent, however if you separate the difference into two sides you work with two divergent series.
Remember \( \lim_{n\to\infty} a_n - b_n = \lim_{n\to\infty} a_n - \lim_{n\to\infty} b_n \) *only* if all (or at least two of the three) limits exists.
