Answer
$$\rho = \frac 12$$
Work Step by Step
1. Use the ratio test to test for convergence:
$$\lim_{n \longrightarrow \infty} \Bigg| \frac{ \frac{(2x+1)^{n+1}}{(n+1)^2} }{ \frac{ (2x + 1)^n}{n^2} } \Bigg | = \lim_{n \longrightarrow \infty} \Bigg| (2x + 1) \frac{n^2}{(n+1)^2} \Bigg| = 2x +1$$
- Therefore, the series converges absolutely when $2x + 1\lt 1$, or $x + \frac 12 \lt \frac{1}{2}$
$$\rho = \frac 12$$
