use the ratio test to find the radius of convergence, we have lim(n→∞) |(a_n + 1)/a_n| = lim(n→∞) |(-7·3^(n…

Explanation:

Step1: Identify $a_n$ and $a_{n + 1}$

Given the form of the ratio - test setup, if $a_{n+1}=-7\cdot3^{n + 1}(x + 3)$ and $a_n=-7\cdot3^{n}(x + 3)^{n}$.

Step2: Simplify the ratio $\left|\frac{a_{n+1}}{a_n}\right|$

[ \begin{align*} \lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}}{a_n}\right|&=\lim_{n\rightarrow\infty}\left|\frac{-7\cdot3^{n+1}(x + 3)}{-7\cdot3^{n}(x + 3)^{n}}\right|\ &=\lim_{n\rightarrow\infty}\left|\frac{3^{n}\cdot3\cdot(x + 3)}{3^{n}(x + 3)^{n}}\right|\ &=\lim_{n\rightarrow\infty}\left|\frac{3}{(x + 3)^{n - 1}}\right| \end{align*} ]

Answer:

$\lim_{n\rightarrow\infty}\left|\frac{-7\cdot3^{n+1}(x + 3)}{-7\cdot3^{n}(x + 3)^{n}}\right|=\lim_{n\rightarrow\infty}\left|\frac{3}{(x + 3)^{n - 1}}\right|$