Will values assigned to divergent series match a taylor series past the radius of convergence? With what I've seen in nearly every case this is true but there are some cases where the function goes to infinity. I'm thinking specifically of the following centered at $0$ and evaluated at $1$.
$y=ln(x-1) \to 1 + 1/2 + 1/3 + ... = \gamma$
$y=1/(x-1) \to 1 + 1 + 1...=-1/2$
$y=(x-1)^2 \to 1 + 2 + 3 + ... = -1/12$
Are these different because of the summability methods that assign their values, or because the value of the function mapped by the taylor polynomial is infinite at that value of $x$, or something else entirely? Are there any finite valued functions which differ from the given summability method past their tayler series' radius of convergence?