The given series is: $$\sum^\infty_{n=0}\left(x^{2n+1}+2x^{2n+2}\right)$$ I know that the interval of convergence can be found by the ratio test, so what I tried was: $$\lim_{n \to \infty}\frac{x^{2n+3}+2x^{2n+4}}{x^{2n+1}+2x^{2n+2}}=x^2$$ Now we know that $-1<|x^2|<1$ or just $x<1$.
Then solving for the single endpoint: $$\sum^\infty_{n=0}\left(1^{2n+1}+2^{2n+2}\right)$$ $$\lim_{n\to\infty}1^{2n+1}+2^{2n+2}=\infty$$ So the endpoint diverges, and the interval is $(-\infty,1)$
The series does not converge for $x \leq -1$. A series $\sum a_n$ cannot converge unless $a_n \to 0$. Can you check that in this case the general term does not tend to $0$ if $ x \leq -1$? The correct interval of convergence is $(-1,1)$.