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 What is the convergence radius of this power series? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/27/2011, 09:00 AM (This post was last modified: 05/27/2011, 09:02 AM by bo198214.) I am really wondering whether one can something achieve with fractional differentiation with respect to tetration, sounds quite promising, however (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. « Next Oldest | Next Newest »