1. ## Radius of Convergence at a point?

Say I have a power series f(z) on the complex plane. Now I know what it means to find the radius of convergence, I know how to use the test, I know how to take the limit superior and use the formula. But what does it mean when one ask what is the radius of convergence of a power series at a particular point z, say z = 1+i?

Thank you.

2. The taylor expansion of a complex $f(*)$ 'somewhere around' a point $z_{0}$ is...

$f(z)= \sum_{n=0}^{\infty}\frac{f^{n}(z_{0})}{n!}\cdot (z-z_{0})^{n}$ (1)

In general is $z_{0}= |z_{0}|\cdot e^{i\theta}$. If $z_{0}$ is not on real axis you can use the substitution $w=z\cdot e^{-i\theta}$ so that the series expansion (1) has only real terms. In particular the radious of cenvergence of the series for $f(z)$ and $f(w)$ is the same...

Kind regards

$\chi$ $\sigma$

3. So if a power series converges everywhere at a point say 0, then does that also means it converges everywhere at other points as well?

4. If a power series converges 'everyhere around' a point $z_{0}$, that means by definition that it coverges 'everywhere around' any other point which is not $z_{0}$... obvious ...

Kind regards

$\chi$ $\sigma$