Does there exist a complex numbers serie $a_n$ (n is natural number) that $\sum_{n=1}^
∞ a_nz^n$ converge when $z=1+i$ and $z=100+40i$
and diverge when $z = 40+80i$ . Any ideas how to show this?
2026-03-26 06:26:23.1774506383
Complex numbers series
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No, because if a power series centered at $0$ converges at $100+40i$ then it also converges at any $z\in\mathbb C$ such that $|z|<|100+40i|$. And $|40+80i|<|100+40i|$.