Cauchy Hadamard formula and starting index of power series

Question

The radius of convergence $r$ can be calculated for every power series $\sum_{k=0}^\infty a_k z^k$ with $a_k\in \mathbb C$ and $z\in \mathbb C$ by using the Cauchy Hadamard formula:

$r = \limsup_{k\to\infty} |\frac{1}{a_k}|^{1/k}$ .

In every textbook I find the power series starting from $k=0$.

Can the Cauchy Hadamard formula also be used for a power series $\sum_{k=1}^\infty a_k z^k$ directly (ie. without changing the index of the power series before calculating $r$)?

score 3 · Accepted Answer · answered Oct 04 '18 at 10:22

3

Certainly. The two series converge for the same points $z$ and have the same radii of convergence. You can thing of the second as the first with $a_0=0$.

answered Oct 04 '18 at 10:22

Kavi Rama Murthy

311,013

Ok that makes it clear. Many thanks. – Carlos Oct 04 '18 at 10:24

score 2 · Answer 2 · answered Oct 04 '18 at 10:22

2

Yes, of course. One term doesn't have any effect on convergence of a series.

answered Oct 04 '18 at 10:22

Mark

39,605

Cauchy Hadamard formula and starting index of power series

2 Answers2