1. radius and interval of convergence

could someone help me with this problem? stuck on homework added the print screen of the problem

2. Originally Posted by pham07
could someone help me with this problem? stuck on homework added the print screen of the problem

$a_n=\frac{n(x-2)^n}{n^2+1}\Longrightarrow\left|\frac{a_{n+1}}{a_ n}\right|=\left|x-2\right|\frac{n}{n+1}\frac{(n+1)^2+1}{n^2+1}$ $\xrightarrow [n\to\infty]{}|x-2|<1\Longleftrightarrow 1

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

3. alright thanks.. I did it by myself and this is what I got.. did I do it correctly?

4. Originally Posted by pham07
alright thanks.. I did it by myself and this is what I got.. did I do it correctly?

Well...almost, if I understood what you meant with those weird arrows:

$\left|\frac{a_{n+1}}{a_n}\right|\xrightarrow[n\to\infty]{}\frac{1}{4}|x+1|<1\Longleftrightarrow |x+1|<4\Longleftrightarrow -5 and etc.

Tonio