Which of the following facts are true of a sequence satisfying $\lim a_n^{\frac{1}{n}}=1$? 
Let $a_n$ be a sequence of non-negative numbers such that
$$\lim a_n^{\frac{1}{n}}=1$$
Which of the following are correct?

*

*$\sum a_n$ converges

*$\sum a_nx^n$ converges uniformly on $[-\frac{1}{2},\frac{1}{2}]$

*$\sum a_nx^n$ converges uniformly on $[-1,1]$

*$\lim \sup\frac{a_{n+1}}{a_n}=1$

My effort:
1.false ,consider $a_n=n$
2.true,The radius of convergence of  a power series $\frac{1}{R}=\lim a_n^{\frac{1}{n}}=1\implies R=1$ .Hence the series converges uniformly for compact sets inside $|x|<1$.Hence the series converges uniformly in $|x|\leq 0.5$
3.false ,putting $x=1$ the same as in case 1.
4.I am unable to prove this fact.How to solve this .

Your answer to 2) is wrong. $R=1$ implies the power series converges absolutely for $|x|<1,$ but not uniformly in that range. Example: $\sum x^n.$ However, the power series will converge uniformly in $[-a,a]$ for all $a\in [0,1).$
