Question: What can we conclude about the convergence and uniform convergence of an infinite series, if it's radius of convergence is infinite ?
For example: Let us consider an infinite series $~\sum_{n=1}^\infty (x/n)^n~$.
We have to conclude the decision for $x\in(-\pi,\pi)$.
Radius of convergence of the series is $$\dfrac 1R=\lim_{n\to\infty}\dfrac{n^n}{(n+1)^{n+1}}=\lim_{n\to\infty}\dfrac{1}{(1+1/n)^n}\cdot\dfrac{1}{n+1}=0\implies R=\infty$$
So here radius of convergence is infinite.
Now wikipedia says, "The radius of convergence is infinite if the series converges for all complex numbers $z$." So following this statement we can say that the series is convergent.
Is it okay ? What about the uniform convergence ?
For a power series, let $I$ be the interior of the domain of convergence. Then the series converges uniformly on compact subsets of $I$. This is true even when the domain of convergence is the whole space.