Well I am not completely sure what you mean with recentering. But is that still applicable with my previous explanations?
Anyway here are the root tests (however only until 100, because creating the Carleman matrix at non-0 is unexpectedly time consumptive) \( |a_n|^{-1/n} \)
islog_e developed at 0 and 1, root test:
isexp_e developed at 0 and 1, root test:
The coefficients of the development at 1 are of course not the coefficients of the Abel function itself but the coefficients of \( \alpha \) which is the Abel function of \( g(x)=e^{x+1}-1 \) in my previous post.
Edit: Ah now I get what you mean with recentering. If you have already a truncated powerseries and want to know the powerseries development at a different point then of course this different point must lie inside the convergence radius of the powerseries. But see Jay here it is different I dont recenter a (truncated) powerseries, but a function and *then* I compute its powerseries. I can do that at any point of the function without regarding convergence and convergence radiuses.
Anyway here are the root tests (however only until 100, because creating the Carleman matrix at non-0 is unexpectedly time consumptive) \( |a_n|^{-1/n} \)
islog_e developed at 0 and 1, root test:
isexp_e developed at 0 and 1, root test:
The coefficients of the development at 1 are of course not the coefficients of the Abel function itself but the coefficients of \( \alpha \) which is the Abel function of \( g(x)=e^{x+1}-1 \) in my previous post.
Edit: Ah now I get what you mean with recentering. If you have already a truncated powerseries and want to know the powerseries development at a different point then of course this different point must lie inside the convergence radius of the powerseries. But see Jay here it is different I dont recenter a (truncated) powerseries, but a function and *then* I compute its powerseries. I can do that at any point of the function without regarding convergence and convergence radiuses.