Answer
The interval of convergence for the series is: $[-2,0]$ and the radius of convergence is $R=1$.
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$ 1. For $l \lt 1$, the series is absolutely convergent. 2. For $l \gt 1$, the series is divergent. 3. For $l = 1$, the series is inconclusive.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|\\=\lim\limits_{k \to \infty} \dfrac{(x+1)^{2k+3}}{(k+1)^2+4} \times \dfrac{k^2+4}{(x+1)^{2k+1}} \\=\lim\limits_{k \to \infty} \dfrac{(x+1)^2(k^2+4)}{k^2+2k+5}\\=(x+1)^2$
So, we can conclude that the given series converges absolutely if $l=(x+1)^2 \lt 1$ or, $-1 \lt (x+1) \lt 1\implies -2 \lt x\lt 0 $ and the test is inconclusive if $(x+1)^2=1$ for $x=-2$ or, $x=0$ and diverges if $l=(x+1)^2 \gt 1$ .
Therefore, the interval of convergence for the series is: $[-2,0]$ and the radius of convergence is $R=1$.
