Radius of convergence, four examples

1. $\left|\frac{a_{n+1}}{a_n}\right|=n+1\to\infty$, so $R=0$: converges only at $x=0$.

2. $\left|\frac{a_{n+1}}{a_n}\right|=\frac1{n+1}\to0$, so $R=+\infty$: converges for all $x$.

3. $\left|\frac{a_{n+1}}{a_n}\right|=\frac n{n+1}\to1$, so $R=1$. At $x=1$: harmonic series, diverges. At $x=-1$: alternating harmonic, converges. Interval of convergence: $[-1,1)$.

4. Also $R=1$. At $x=\pm1$: $\sum\frac1{n^2}$ and $\sum\frac{(-1)^n}{n^2}$ both converge. Interval of convergence: $[-1,1]$.

Remark. The endpoints are the delicate part: inside the open interval the convergence is absolute, outside the series diverges, but on the boundary anything can happen.