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 .

First of all, way to go for your efforts. As far as I can see, your answers to the first three questions are correct. To refute the last one consider the sequence
$$\{a_n\}=\{1,1,2,1,3,1,4,1,5,1,6,1,....\}=\begin{cases}k,&n=2k-1\\{}\\1,&\text{otherwise}\end{cases}$$
Observe that
$$\left\{\frac{a_{n+1}}{a_n}\right\}=\left\{1,2,\frac12,3,\frac13,4,...\right\}$$ 
