Given the Laurent series $$\sum_{n= - \infty}^{\infty} \frac{z^n}{3^n + 1}$$ Find the annular region for which it converges. I'm struggling to find any similar examples or where to begin for this.
2026-03-31 20:09:00.1774987740
Annular regions for which this Laurent series converges
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Following the excellent comment by G. Sassatelli (fill in details):
$$\sum_{n=-\infty}^\infty\frac{z^n}{3^n+1}=\sum_{n=1}^\infty\frac{3^n}{(3^n+1)z^n}+\sum_{n=0}^\infty\frac{z^n}{3^n+1}$$
For the first sum above, we have
$$\left|\frac{\frac{3^{n+1}}{(3^{n+1}+1)z^{n+1}}}{\frac{3^n}{(3^n+1)z^n}}\right|=\frac{3(3^n+1)}{\left(3^{n+1}+1\right)|z|}\xrightarrow[n\to\infty]{}|z|^{-1}$$
For the second one:
$$\left|\frac{\frac{z^{n+1}}{3^{n+1}+1}}{\frac{z^n}{3^n+1}}\right|=\frac{(3^n+1)|z|}{3^{n+1}+1}\xrightarrow[n\to\infty]{}\frac{|z|}3$$