Cn = 1/(n^2+1)

C(n+1) = 1/[(n+1)^2+1] = 1/(n^2 +2n +2)

lim|(x-5)^(n+1)C(n+1)/[(x-5)^nCn ]|< 1

|x-5| lim (n^2+1)/(n^2 + 2n +1) < 1

lim (n^2+1)/(n^2 + 2n +1) = 1 Use L'hopital's Rule on the corresponding continuous function if you need or note the lim is just the limit of the highest powered terms.

so the radius of convergence is 1

|x-5| < 1

We have convergence for 4 < x < 6

Check x= 4 and x =6 in the original series to determine the convergence issue at the endpoints