$\displaystyle a_n=\frac{n(x-2)^n}{n^2+1}\Longrightarrow\left|\frac{a_{n+1}}{a_ n}\right|=\left|x-2\right|\frac{n}{n+1}\frac{(n+1)^2+1}{n^2+1}$ $\displaystyle \xrightarrow [n\to\infty]{}|x-2|<1\Longleftrightarrow 1<x<3$
From here you get the radius of convergence and almost the interval of convergence: you just have to check whether the extreme points 1, 3 belong to it or not.
Now you do the second one by yourself.
Tonio