Answer
$\{a_{n}\}$ diverges.
Work Step by Step
$a_{n}=\displaystyle \frac{1}{(0.9)^{n}}=\left( \frac{1}{0.9}\right)^{n}=\left( \frac{1}{\frac{9}{10}}\right)^{n}=\left( \frac{10}{9}\right)^{n}$
The function $f(x)=a^{x},\ a\gt 1$
increases without bound when $ x\rightarrow\infty$.
Thus, $\displaystyle \lim_{n\rightarrow\infty}\{a_{n}\}$ =$\infty.$
$\{a_{n}\}$ diverges.
