# Radius of convergence of a complex series

I need to find the radius of convergence of the following series $$\displaystyle \sum_{n=0}^{\infty} \Big(\frac{z}{1+z}\Big)^n.$$

Here is my solution:

I know that $$\displaystyle \sum_{n=0}^{\infty} w^n$$ converges absolutely for $$|w| < 1$$, converges uniformly for $$|w| \leq \delta < 1$$ and diverges elsewhere. Applying this to the above problem, the given series converges absolutely iff $$\Bigg| \frac{z}{1+z} \Bigg| < 1 \iff \text{Re}(z) > \frac{-1}{2}$$

Hence, $$\displaystyle \sum_{n=0}^{\infty} \Big(\frac{z}{1+z}\Big)^n$$ converges absolutely for $$|z|< \dfrac{1}{2}$$ and uniformly on $$\Big(|z| < \dfrac{1}{2}\Big)\setminus \Big\{\dfrac{-1}{2}\Big\}$$

• How do you get uniform convergence on $(|z| <\frac 1 2)\setminus \{\frac {-1} 2\}$? – Kavi Rama Murthy Jan 18 '20 at 5:46
• Sorry. It is not. Now, I feel it is absolutely convergent on $\Big(|z| \leq \frac{1}{2} \Big) \setminus \Big\{\frac{-1}{2}\Big\}$ and uniformly convergent on $\Big(|z| \leq \frac{1}{2} \Big) \setminus \Big\{\text{small arc near } z = \frac{-1}{2}\Big \}$, since it is compactly contained inside the set $\text{Re}(z) > \frac{-1}{2}$. – Ajay Kumar Nair Jan 18 '20 at 6:12

Your main error is assuming that the area of convergence is a disk (with the border being 'unknown'), by insisting on a convergence radius!

That's only true for a power series, which has the form $$\sum_{n=0}^{\infty}a_nz^n$$, which your series is not. Other kinds of serieses have other shapes of their area of (absolute) convergence. As you correctly found out, your series has a half plane as area of absolute convergence:

$$\Re(z) > -\frac12$$

You then tried to interpret that result, saw the value $$\frac12$$ and tried to cram into your (wrong) notion that a convergance radius must exist. But that's not true, the area of absolute convergence is a half plane, not some disk!

Uniform convergence is another matter. Aside from solving

$$\left\vert\frac{z}{1+z}\right\vert \le \delta$$

for a given $$0 < \delta < 1$$, you need to find out if the map

$$z \to \frac{z}{1+z}$$

is uniformly continuous in that area as well.