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Radius of convergence

I'm asked to expand

\dfrac{1}{1-z}

around

\omega

, where

\omega^3 = 1, \ \omega \not= 1

.

I've found the series to be

\displaystyle\sum_{n=0}^{\infty} \dfrac{(z-\omega)^n}{(1-\omega)^{n+1}}

which is valid for

\dfrac{(z-\omega)}{(1-\omega)} < 1 \implies | z-\omega |< | 1-\omega |

.

Now... applying the ratio test, I find that

Unparseable latex formula:

\displaystyle\lim_{n\to \infty} \vline \dfrac{a_{N+1}}{a_N} \vline = \displaystyle\lim_{n\to \infty}\left(\dfrac{1}{\vline 1 - \omega \vline} \lvert z - \omega\rvert\right)

What exactly does this imply? Does this mean that the series converges when

|z|<w

Reply 1

SimonM

I don't understand why you've applied the ratio test. You've worked out where the series is valid for.

The radius of convergence is |1-w|. The series converges for |z-w|<R, where R is the radius of convergence, and you know R!