As we know, for the power series $$\sum_{n=1}^{\infty}a_{n}(x-x_0)^n,$$ it will have a convergence radius $R$ and $R$ is non-negative. I want to know whether there exists a power series example that its convergence domain is $[-R,R]$ (that is a closed domain) and in particular, the power series on $x=R$ and $x=-R$ are both conditionally convergent?
It’s easy to find an example that power series on $x=R$ and $x=-R$ are absolutely convergent. But it is difficult to find the (both $R$ and $-R$) convergent case.
Take, for instance,$$\sum_{n=1}^\infty\frac{(-1)^n}nx^{2n}.$$Its radius of convergence is $1$ and it converges conditionally when $x=\pm1$.