Seja $a_n = \displaystyle\frac{2^{n}(x+1)^{n}}{n^2}$. Aplicando o teste da razão:
$ \left| \displaystyle\frac{ a_{n+1}}{a_n}\right| = \left| \displaystyle\frac{ 2^{n+1}(x+1)^{n+1}}{ (n+1)^2}\cdot \displaystyle\frac{n^2}{2^{n}(x+1)^{n}}\right|= \left| \displaystyle\frac{2^{n+1}}{2^{n}}\cdot \displaystyle\frac{(x+1)^{n+1}}{(x+1)^{n}}\cdot \displaystyle\frac{n^2}{(n+1)^2}\right| = 2|x+1|\left(\displaystyle\frac{n}{n+1}\right)^{2}= 2|x+1|\left(\frac{1}{1+\frac{1}{n}}\right)^2 \Rightarrow $
$ 2|x+1|<1 \Leftrightarrow |x+1|<\frac{1}{2} \Leftrightarrow -\frac{1}{2} < x+1< \frac{1}{2} \Leftrightarrow -1-\frac{1}{2} < x < -1+\frac{1}{2} \Leftrightarrow -\frac{3}{2} < x < -\frac{1}{2} $ .
Logo, os extremos do intervalo são: $ -\frac{3}{2} $ e $ -\frac{1}{2}$.
