Find Taylor's polynomial of the following function in : .

Determine the radius of convergence of the series.

My attempt :

. Which answers the first question.

Now trying to figure out the second question :

So .

I apply the ratio test : which tends to when tends to .

Hence the radius of convergence is ...? It makes no sense at all.

I'd like to know where is(are) my error(s). Thanks a lot!