Verified answer

Use the ratio test.

r = lim(n→∞) |[(x + 3)^(n+1) / (n+1)^2] / [(x + 3)^n / n^2]|

..= |x + 3| * lim(n→∞) [n/(n+1)]^2

..= |x + 3| * lim(n→∞) [1/(1 + 1/n)]^2

..= |x + 3|.

This series converges (at least) for r = |x + 3| < 1.

Checking the endpoints:

x = -2 ==> Convergent p-series (with p = 2).

x = -4 ==> Absolutely convergent (by previous remark).

So, the series converges iff x is in [-4, -2].

I hope this helps!

Try the Ratio Test.

lim | 1/(n+1)^2 * (x+3)^n+1 / (1/n^2 * (x+3)^n) |

lim | (x+3) * (n^2/(n+1)^2) |

|x+3| lim |n^2 / (n+1)^2|

|x+3| * 1

|x+3|.

So, as long as |x+3| < 1, the series converges. Therefore, the radius of convergence R = 1.