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)^{k+1}}{(k+1)} \times \dfrac{k}{(x+1)^k} \\=\lim\limits_{k \to \infty} \dfrac{|x+1|k}{k+1}\\=|x+1|$
So, we can conclude that the given series converges absolutely if $l=|x+1| \lt 1$ or $-1 \lt (x+1) \lt 1\implies -2 \lt x\lt 0 $. The test is inconclusive if $|x+1| \gt 1$ for $x=-2$ or $x=0$. It diverges if $l=|x+1| \gt 1$.
Therefore, the interval of convergence for the series is: $(-2, 0]$ and the radius of convergence is $R=1$.
