Answer
The interval of convergence for the series is: $(\dfrac{-19}{3}, \dfrac{-11}{3})$ and the radius of convergence is $R=\dfrac{4}{3}$.
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{3}{4} |x+5|\\=\dfrac{3}{4} |x+5|$
So, we can conclude that the given series converges absolutely if $|x+5| \lt \dfrac{4}{3}$ and diverges if $|x+5| \gt \dfrac{4}{3}$ .
Therefore, the interval of convergence for the series is: $(\dfrac{-19}{3}, \dfrac{-11}{3})$ and the radius of convergence is $R=\dfrac{4}{3}$.