I am trying to determine the convergence of $\sum_{n=1}^{\infty} \frac{-1^{n}*(n)^2*(x+2)^n}{3^n}$. When I do the ratio test I get the limit of $\frac{(-1)*(x+2)}{3}$, or $\frac{-1}{3}*|x+2| < 1$, however the answer is that the radius of convergence is p = 3. Did I incorrectly compute the limit? Any guidance would be very appreciated, thanks!
2026-04-02 17:03:08.1775149388
Ratio test for complicated functions
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You have two problems. When you say $\frac{-1}{3}*|x+2| < 1$, this is true for any $x$ as the left side is negative. If you are taking absolute values you should include the $-1$ in the absolute value, which means you can ignore it. Now you have $\frac{1}{3}*|x+2| < 1, |x+2| \lt 3, x \in (-5,1)$ and the radius of convergence is indeed $3$. The interval is centered on $-2$.